New Functional Techniques and Methods of Path Integration
by SCOTT B ANDERSON
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Physics
Montana State University
© Copyright by SCOTT B ANDERSON (1984)
Abstract:
We develop three new functional techniques. The first is the method of delta functionals whereby a
path integral having a Hamiltonian linear in position is reduced to quadratures through the evaluation
of an equation of evolution. We augment this technique through the introduction of three canonical
transformations which may be used to simplify the path integral. We also construct a new path integral
from coherent states for a time—dependent harmonic oscillator. Finally we solve a new quantum
mechanical problem and introduce the concept of a functional anti-derivative. NEW FUNCTIONAL TECHNIQUES AND METHODS
OF PATH INTEGRATION
by
Scott Buckingham Anderson
A thesis submitted in partial fulfillment
of the requirements for the degree
of .
Doctor, of Philosophy
in
Physic s
MONTANA STATE UNIVERSITY
Bozeman, Montana
June 1984
APPROVAL
of a thesis submitted by
Scott Buckingham Anderson
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College of Graduate Studies.
Graduate Committee
Approved for the Majo
Date
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Date
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iii
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TABLE OF CONTENTS
Page
ABSTRACT....... ........................... . ...... .
v
1. BASIC CONCEPTS. ..........
I
2. METHODS OF PATH INTEGRATION.........
12
3. THE METHOD OF DELTA FUNCTIONALS. ......
20
4. CANONICAL TRANSFORMATIONS ........______
26
Linear Momentum Transformation.................. ..
Point Canonical Rescaling.............
Time Rescaling.........
26
29
35
5. RELATIVISTIC PROPAGATORS........... ...... ____ ____ ...
41
6 . GROUP MANIFOLDS.......... ... ..........................
45
7. TIME-DEPENDENT COHERENT STATE PATH INTEGRALS__ _____
52
Time— dependent Coherent States....................
Construction of the PatA Integral......... .
53
57
8 . NEW SOLVABLE QUANTUM SYSTEM..........................
61
9. STRUCTURE OF THE GENERATING FUNCTIONAL........
69
10. SUMMARY AND SPECULATION........ ...............
78
REFERENCES CITED..........
82
APPENDICES...... ............... . . ____ _______ ___________
86
Appendix
Appendix
Appendix
Appendix
A — Time— slicing Derivations.............
B — Formulae and Integrals.............. .
G — Linear Momentum Transformations......
D — Time— dependent Harmonic Oscillator...
87
94
98
100
V
ABSTRACT
We develop three new functional techniques.
The first
is the method of delta functionals whereby a path integral
having a Hamiltonian linear in position is reduced to quad­
ratures through the evaluation of an equation of evolution.
We augment this technique through the introduction of three
canonical transformations which may be used to simplify the
path integral.
We also construct a new path integral from
coherent states for a time— dependent harmonic oscillator.
Finally we solve a new quantum mechanical problem and intro­
duce the concept of a functional anti-derivative.
I
CHAPTER I
BASIC CONCEPTS
This thesis is devoted to the development of new
functional methods.
The primary emphasis is upon the
explicit calculation of path integrals which are functional
integral representations of the Green's functions for the
Schroedinger equation.
The author first provides a review
of the basic concepts which are standard and are concisely
reviewed by Marinov (1980).
The history of path integration
is also discussed towards the end of the chapter.
involve the reader in a participatory manner,
To
the thesis is
written in the first person plural.
A word about conventions and notation.
units where h = c = I.
We use natural
In many calculations we will also
take the mass and frequency to be unity.
Vectors in three
dimensions will be boldface and 4—vectors will simply be in
lower case.
eg: a.b.
Scalar products will be denoted by a period,
The product symbol, usually denoted by an upper­
case n , will be taken to be II.
the Minkowski tensor is
Our metric convention for
and Greek letters will be
used for tensor indices in spacetime.
with argument x as usual: f(x).
We write a function f
We write a functional g
2
with argument F(t) as g [F] .
Remember that a functional
number that depends upon a function for its value
is a
(eg: the
area under a curve, A[F] = / F(x) dx).
We start with the Schroedinger equation for the wave
function vH in one dimension
HT=
idtT
(1 .1 )
with initial condition V (0,x)„
Since it is a linear
partial differential equation we may solve it in terms of
the initial wave function vT (0 ,x) if we can construct a
Green's function K that satisfies
(8t + iH(p,x))K(x,tla,f) = 5(x-a)8(t-r).
(1.2)
If this can be done we may then write tT (x,t) as
V
(x,t) = / da K (x,tI a ,O ) V (a ,O )
(1.3)
which requires the limit for K
limit t — > r, K(x,tla,r) — > 8 (x-a).
(1.4)
We will also take K(x,t|a,r) = O for t < r which may be
enforced by the introduction of the factor G(t-r), the unit
step function.
We also note that K itself satisfies Eq.
(1.3)
K (c,tI a ,r) = / db K (c ,tIb,s)K (b,s Ia ,r ) .
which is the defining property of a semigroup.
(1.5)
Because K
has this quality of propagating the wave function and itself
forward into the future it is often called the propagator
and that nomenclature will be used in this thesis.
Before we show how K may be represented by a functional
integral we discuss some of its properties which make it a
3
useful quantity to know.
First we note that K has a
standard construction in terms of the energy eigenfunctions
vVg(X) with T = s - r,
K(b,s|a,r) =
I e"iETvV E (b)SJ "'gCa) .
(1.6)
A useful function constructed from the propagator is the
spectral function Y(T) which is defined as
Y(T) = / da K (a ,T Ia .0)
(1.7)
and by using E q . (1.6) we easily see that
Y(T) =
I e“ iET.
(1.8)
Information on the ground state energy may be found by
taking the limit T — > -i® of Y(T) since only the exp(-IEq T)
term will survive to contribute to the sum.
The Fourier transform of K(T)
Writing
is also of interest.
(we suppress the spatial dependence)
K (b)) = Z d T
K(T) e iti)T
(1.9)
and once again using E q . (1.6) we find that
KU)
= i
I vV E (b) <V * E (a) (to - E )-1
which reveals that K U )
(1.10)
has poles at bound state energies
and a cut along the continuum.
Another quantity of interest that may be calculated
from the propagator is the generating functional Z [F ]„
It
is found from the propagator K[F](bI a ) where a driving term
F (t )x has been added to the Lagrangian.
Specifically,
if
vV 0 (X) is the ground state wave function for the undriven
system then
Z tF ] = N /
dbda V *0 <b )
(a ) K [F ] (b , s Ia ,r ) .
(1 .1 1 )
4
We require that the source F (t) be turned off at the
endpoints F (s) = F (r) = 0 and determine the normalization
constant N through the boundary condition Z [03 = I.
Z [F ] is of enormous importance as it is ( i n q u a n t u m
field theory) the generating functional for the N-point
Green's functions.
They may be found by taking N functional
derivatives of Z [F 3 and then setting the source F (t ) to
zero.
A functional derivative is somewhat like a partial
derivative.
We give some examples:
8
F (s ) = 8 (t-s ) ,
8F(t)
8F(t)
f F (s )G( s ) ds = G (t )
— --- exp{// dxdy F (x)G(x,y)F (y)} = i f dx G(t,x)F(x).
SF (t )
We now wish to discuss the r e p r esentation of K b y a
functional integral.
This was first done in the context of
quantum mechanics by Feynman (1948) in the early Forties
based upon an observation by Dirac that the form for the
infinitesimal propagator K(At) is approximately the
exponential of the classical action.
Feynman took this form
as axiomatic and succeeded in constructing all of quantum
mechanics from his functional integral representation of the
propagator K.
For a very interesting anecdotal recounting
of those halcyon days we recommend reading Feynman's (1972)
Nobel lecture.
We will take a different route and derive K
as a functional integral from standard quantum theory.
The
5
derivation that we follow is the route taken by Abers and
Lee (1973) ,
To begin we write K in the Schroe dinger representation
as
K(b,s I a „r) = <b|e“ iHTla>.
(1 .12 )
Then we insert N-I resolutions of the identity
2 Ix > <x I = I
to write K as
K -
I <1 le"iHAt !=„_!> <^N-i I="11111 U N_2> x...
x ... x <X2 I e- *®A t lx j> <xj I e-
a>
(1.13)
where we have defined At = tn - tn„^ and NAt = T = s - r .
For convenience we will also define xjj = b and xQ = a.
Now we examine the infinitesimal propagator K(At) =
<xn I e
We may insert a complete set of momentum
eigenstates
K(At)=
Ip > to write K(At) as
;
2 <xn lpn_1/2><pn_1/2 lxn_ 1 >
x exp{-iAtH(pn_1/2 ,xn ,xn_ 1 )} .
(1.14)
We will discuss the explicit form of the Hamiltonian in a
moment in connection with the factor-ordering problem.
Using the explicit form for <x Ip > allows us to rewrite K(At)
as
K(At) = f dpn_1/2 (2n)-1 exp(ipn_ 1/2 (xn - Xj^ 1 )) (1.15)
x exp{— iAt(H(pn_1/2 ,xn ) + H(pn_1/2 ,xn_ 1 ))/2} .
The choice for the Hamiltonian in the second term on the
right is somewhat ambiguous and is related to the factorordering problem of quantum mechanics.
The usual form taken
is H(pn-^ 2 , (xn + xn_^ ) / 2 ) but we eschew the conventional
6
wisdom and write H in the Euler approximation form as it
will allow us to deduce the precise ordering of operators in
the Schroedinger equation that K satisfies.
_y
Because all the K(At) 's are exponential in form we may
concatenate them to write K(T) as
K(T) = II^"1 f dxn II^ / dpn_1/2 /2 jt
N
x e x p {i S 1 Pn-I /2 ^xn - xn-l)}
x exp{-i
(1.16)
(H(Pn_i/2 .=n) + H(^n-I/2 'xn-l> >AtZ21
or symbolically as
K = / [dx dp / 2tt] e xp (i/ px - H( p ,x ) dt).
(1.17 )
The above integral is to be interpreted as a functional
integral where one integrates over functions rather than
points as in a conventional Riemann integral.
In particular
we are instructed in the integral for K to integrate over
all momentum functions p(t) and over all position functions
x (t) that start at x (O) = a and end at x(T) = b .
certain
tion.
sense
we have
'integrated'
In a
the Schroe dinger
equa­
The question that comes to mind of course is how
on
earth would one ever evaluate an integral over functions and
that is one of the main topics of this thesis.
We would now like to make some general remarks about
C
the functional
integral for K.
We notice that the exponent
in the integral is just the action S of classical mechanics
in Hamiltonian form.
This suggests an interpretation for K.
A particle may travel from point a to point b via many
paths.
Classically only one path is selected,
the one that
7
has the minimal amount of action.
mechanics
However in quantum
(QM) anything that can happen will happen with a
certain amplitude.
The prescription in QM is that the total
amplitude for an event is the sum of the amplitudes for all
possible ways the event can occur.
The amplitude for a
particle going from a to b along a certain path is
exp{iS[path]} and K(b,s|a,r), the total amplitude for the
particle to go from a to b , is just the sum (or integral) of
the amplitudes for all possible paths.
Hence the termi­
nology of the path integral.
We now turn to the question of what is the equation
solved by K?
A good discussion of this question for our
time-slicing procedure is given in the paper by Mayes and
Dowker (1973).
functional
In the first appendix we show that the
integral represents a solution to the specific
Schroe dinger equation
i"
K (q, t I a ,r) = H (p,q) K (q, t I a ,r )
(1.18)
a
where the quantum operator p is represented as p = — i—
.
0q
The subscript S on the Hamiltonian stands for a very
specific ordering of the operators p and q.
We represent
this ordering by writing Hg(p,q) = -(H(p|q) + H ( q Ip)) where
H(p|q)
is a right ordered function of q which means that in
a power series expansion of H (p,q)
nm a11
H(p.q) = 2 Hffln - - - all of the quantum q ’s are placed to the right of the
(1.19)
8
quantum p's, ie:
Pm q11
H(PU) - Z H11
2-
(1.20)
and similarly H (q Ip) is left ordered
,
Q n pm
H(Olp) - 3 H11 3- E- .
(1.21)
A Green's function solution to this Schroedinger
equation may be constructed from the path integral for K by
writing G (b,s Ia ,r) = ©( s-r)K (b,sI a ,r ) with G(t) the unit
step func tion.
G satisfies E q . (1.2)
(1 .2 2 )
(-- + iHg(p,q))G = 8 (q— a)8 (t-r)
We will also be concerned with a generalization of K
namely
K(b, a) = C (b )C (a)-7-7/[ dp/2 ji] [dq]
D (a )
x exp(i f
pq-H(p,q) dt )
(1.23)
I
where C (q) and D (q) are arbitrary functions of position.
This propagator possesses the semigroup property
K (c la) = f A b C(b)“2 K(clb)K(bla)
(1.24)
and has the limit as s -> r that E(bla) -> C (b)2 8 (b-a).
It also satisfies the Schroedinger equation
i~~ K (q,tI a ,r) = Hg(p,q) K (q,tI a ,r)
where the operator p = -iE(q)— E (q) -1 _
dq
with E = CD.
(1.25)
9q
+ i!:w
E (q)
Following Faddeev and Slavnov (19 80, p .26) we may
derive an interesting form for K when
p2
H =
+ V(x).
2
9
Then K is
K = / [dx dp / 2 it] e x p [ i f px - p^/2 -V(x) dt}.
(1.26)
We perform a canonical transformation P = p + x with dP =dp.
K = / [dx dP/2jr] exp{i/ (P + x)x - (P + x)^/2 -V dt}.
The cross terms Px cancel giving
(1.27)
K = (/ [dP / 2 it] exp {-i/ P2/2 dt})
x (/ [dx] exp[i/ x 2/2 - V(x) dt}).
(1.28)
The momentum integrations may be considered a normalization
factor that may be absorbed into the quasi-measure
[dx]
allowing us to write
K = /
[dx] e x p { i / x 2/ 2 - V ( x )
dt}.
(1.29)
We see that the exponent is just the action S written
in Lagrangian form.
This was Feynman's original form for
his version of quantum mechanics.
We note that in general
one must start with E q . (1.15) as the fundamental form for
K.
We will see in chapter 6 that even this form must be
modified should the mechanics take place on other manifolds
than Euclidean space.
The subject of this thesis is the investigation and
development of new methods for dealing with functionals,
in
particular we provide a new technique called the method of
delta functionals
(MDF) for the explicit calculation for a
certain class of path integrals. We extend our technique
through the introduction of three canonical transformations.
Also discussed are the generalizations of MDF to relativity
and group manifolds.
In Chapter 7 we construct a new path
10
integral for time— dependent coherent states.
Finally we
report the results of our investigation of the generating
functional for a new exactly solvable quantum mechanical
system which is the one-dimensional analogue of general
relativity.
In the penultimate chapter we introduce a new
kind of functional integration,
functional anti-differentia­
tion, in connection with the perturbation series for this
new solvable system.
Historically,
the first explicit path integration was
that of the quadratic Lagrangian (harmonic oscillator)
reported in Feynman's
(1948) seminal paper.
The next new
path integral was that of the inverse quadratic potential
calculated in the paper of Peak and Inomata
twenty years later.
(1969) some
Basically the result was obtained from
the observation that a free particle in polar coordinates
has a r- ^ centrifugal potential already.
The propagator for
the Coulomb potential was evaluated by Duru and Kleinert
(1979) by transforming it into an harmonic oscillator in 4
dimensions.
Recently the transformation introduced by Duru
and Kleinert was used by Duru (1983) to obtain the Morse
potential propagator.
This transformation may be used to
obtain other propagators which have not yet appeared in
print.
One sees that the calculation of new propagators has
depended upon the creation of suitable transformations.
Relativistic path integrals were first constructed by
Feynman in his 1948 paper and in Feynman (1951).
Group
11
space propagators were first discussed by Schulman (1968)
and definitively treated by Marinov and Terentyev (1979) .
Coherent states were first used to construct a path integral
by Klauder (1960) .
12
CHAPTER 2
METHODS OF PATH INTEGRATION
In the previous chapter we derived a representation of
the propagator as a functional
integral.
Now we wish to
examine the principal techniques for evaluating functional
integrals in order to be able to compare and contrast them
with our technique which will be presented in the next
chapter.
The example used in all cases will be the harmonic
oscillator (HO) with Lagrangi an L = (x^ - w^x^)/2 and
Hamiltonian H = (p^ + m ^ x ^)/2.
The first method we illustrate hinges upon the fact
that Gaussian or quadratic integrals concatenate, scilicet
the integral of a product of two Gaussians is again a
Gaussian.
This was noticed by Feynman in his first paper on
path integrals.
and Lawande
We follow the derivation given by Ehandekar
(1975).
We first write the Lagrangian (L) path
integral for E in a time-sliced or discretized form
(2.1)
E = A N II / dxn ex p {- 2 (xn - xn_ 1 )2/At - M2Xn2A t } .
A is the normalization constant induced by the momentum
integrations.
Suppose we do the x^, ^2
be left with an integral
xn— I integrals.
We would
13
/ dXn e x p { ~ - ( ( x n+1 - xn )2 - U A t x n+1)2 }
X eXp{2A t ($2na2 + ^nxIi2 " 2PaxH*) )
(2*2)
which upon integration would produce a factor multiplying
the exponential of
eXp{2X t ($2n+ia2 + fl^-Ixn+!2 " 2Pn+l xn+l a) ) •
By using Eq.
(2 •3 )
(B„9) we may obtain the following
recursion relations for £2, p, and P which will allow us to
evaluate the concatenation of Gaussian integrals.
W e define
o = p + I.
an+1 = 2 - M2At2 - l/an .
(2.4)
fin+l ~ Q n ~ Pn2 ^an *
(2.5)
Pn+1 = Pn /an«
(2*6)
To solve the key equation Eq. (2.4) we define Qn+]/Qn =
on .
Then Eq. ,(2.4) becomes
Q n+ 2 - 2 Qn+l + Qn = - » 2A t 2 Qn+ l '
(2 -7)
In the limit as At — > O , Eq. (2.7) reduces to the
differential equation for the harmonic oscillator
d2Q/dt2 + (O2Q = 0.
(2.8)
The solution with the correct boundary condition Q q = 0
is Q(t) =
sin((ot)/(o.
The other functions may be found
from
p (t )/A t = 6 (t )/Q( t ) = (ocot (tot) ,
(2.9)
P (t )/A t = 6Q/Q(t ) = tocsc(tot), and
(2 .10 )
^ (t )/At --- 602/Q(t)2
(2 .11)
with solution Q (t)/A t = (ocot(tot).
(2 .12 )
14
Hence the exponential term resulting from the path
integral is
e xp [-------- ( (b ^ + a^)cos(d)T) - 2 b a)}.
2 sin((oT)
The pre-exponential factors concatenate to
(2.13)
A-N (2Tti) (N-I) /2AtN / 2 (Q(T)/Qo)- l / 2 e
For a convergent functional
(2.14)
integral we must have A = 2TTiAt
which gives as our result for the HO propagator ( T =
s - r)
K(b , s Ia ,r) = (2 tt i s in ((oT)/w)-•*•^ ^
x ex p {--- -— — ((b ^ + a^)cos(wT) — 2ba)} .
(2.15)
2 s in (o)T)
The success of this method is a consequence of the fact
that Gaussian integrals concatenate.
To generalize this
method requires the discovery or creation of other concatenatory integrals involving Gaussians.
This has only been
done in one other instance where the inverse quadratic
propagator was found using E q . (B .11). . Hence we see that
this method has a severe limitation;
a suitable concatenate-
ry integral must be found antecedent to any calculation and
this is a very difficult task.
A second method that is used to evaluate quadratic path
integrals utilizes an expansion about the classical path.
This was also developed by Feynman but the evaluation of the
Fredholm determinant seems to first appear in MontrolI
(1952).
Examine the HO path integral
in Lagrangi an form
K = / [dx]exp{^/ x^ - M^x^ dt }
with boundary conditions x(T) = b and x (O) = a.
(2.16)
Now we
expand x(t) about the classical path q (t ) which satisfies
15
the equation of motion d^q/dt^ + w^q = 0.
Specifically we
write x (t ) = q (t ) + y (t ) and note that dx = dy and y (0) =
y (T ) = 0.
For quadratic Lagrangians
L) we have S[q + y] = S [q] + S [y ] .
(and only for quadratic
Hence we may write K as
K = exp{iS [q]} /[dy]exp{iS[y]}.
(2.17)
We remark that all of the position dependence upon b
and a is contained in the first factor which is simply the
exponential of the classical action Sc (the action evaluated
along the classical path).
This is the familiar
S (b , a ) ----- 7— — ( (b 2 + a2 )c o s (w T ) - 2ba).
0
2 Sin(MT)
(2.18)
The second, path integral factor is a function of T
alone (due to the fact that y (t) vanishes at the endpoints).
We now turn to its evaluation.
Because of the periodic
boundary conditions.we may write the integral as
/[dy] exp {-i/ y (D^ + a>2 )y/2}
(2.19)
where we have performed an integration by parts and defined
D2 = d2/dt2 .
This is a continuum version of E q . (B .5) and
we use it to write E q . (2.19) formally as
C (Be t [D2 + M2] )"1 / 2 .
C is a formally infinite normalization constant.
(2.20)
We have
used the standard notation of capital Det for determinants
of continuous matrices
(Fredholm determinants).
The problem
is now to evaluate the this determinant.
The time— sliced version of this determinant is
A- N (2jTiAt) (N-1 )/2 (detM)-1/2
where the NxN matrix M is
(2.21)
(2 .2 2 )
16
2
-
-I
M2At2
-I
2 - M2At2
O
-I
G
O
-I
O
2 - M 2At2
-I
Using the fact that A must equal 2jtiAt allows us to write
the factor E q . (2.21) as
(2iti)_1/2 (AtdetM)"!/2 .
(2.23)
By inspection of the form for M given above we see that
the quantity f = AtdetM satisfies the recursion relation
fn+l - 2fn + fn_i = -M2At2 fn .
(2.24)
In the limit as At goes to zero this equation reduces to the
differential equation for the HO
D2 f + M2 f = 0 ~
(2.25)
with initial conditions f (0) = 0 and f(0) = I.
The solution
is f (t ) = sin(Mt)/M and consequently upon substitution of
this result into E q .(2.23) and E q . (2.17) we arrive at the
HO propagator given by E q . (2.15).
This method is the principal technique used in the
literature to evaluate path integrals as it is easily
generalized to higher dimensions and field theory.
as should be obvious,
However
this method may only be employed to
calculate Gaussian path integrals and has often led to the
claim that only Gaussian path integrals may be evaluated.
This claim is not true in quantum mechanics but it is
certainly a fact that at present the only path integrals
17
that have been evaluated in quantum field theory are indeed
Gaus sian.
The third meth6d we review exploits the recently
developed formalism of the star product.
A seeming
advantage of path integrals is that they are classical
objects.
space.
One need never work with operators in Hilbert
The star product extends this advantage to such
things as the commutator and the Heisenberg equations of
motion.
The use of the star product allows one to map
operations in Hilbert space to (perhaps simpler) operations
involving ordinary functions.
The concept of the star
product is developed by Bayen et aI . (1978).
Specifically the star product of two phase space
functions f and g is defined as
(f*g)(x,p) = f(x,p)ex p {-(dx dp - 3p dx )}g(x,p).
(2.26)
The quantum mechanical commutator goes over to
[f,g] -~> - i (f*g - g*f)
(2.27)
and in an interesting (but unremarked upon) paper Sharan
(1979) has shown that if we define a star exponential as
(exp*f)(x,p) = 1 + f + f*f/2 ! + ...
(2.28)
then the phase space path integral may be written as
/ [dxdp/2 ti]exp {i/ px - H(p,x) dt }
= /dp / 2 Jt eiP (b" a)
(exp*-iHT) (p,-~-)
.
(2.29)
The obvious difficulty here is the evaluation of the
star exponential.
In the case of a quadratic Hamiltonian
H = (p % + x 2)/2 it may be done as follows.
Let G(t,H) =
18
exp*— iHt.
Next we use the fact that for any function f
H*f(H) = Hf - f '/4 - H f ''/4
(2.30)
to write a partial differential equation for G
H S2G
8G
IdG
(2.31)
" HG " ? ii " 4
Fourier transforming in both variables to a new function
(2.32)
G (s ,to) = f f dH dt/2n e i<dt elsH G (t ,H)
allows us to write a differential equation
(2.33)
i (a) + is/4)G(s,<o) = (I + s 2/4) dG/ ds
which has solution
G (to,s ) = (I + s2/4)-1/f2 exp {2 itotan-1 (s/2) } .
(2.34)
Inverting this result gives
G(t,H)
(2.35)
= se c (t /2)ex p {-2iHta n (t /2)}
and inserting this expression into Sharan's form for the
path integral leads to
K = / dp/,2Jt eip(b-&)
sec(T/2)
x ex p {— ita n (T/2)(p2 + (b+a)2/2)
(2.36)
which upon integration gives again E q . (2.15) with (0 = 1.
One advantage of this method is that it is not limited
to Gaussians.
However the mathematical technology for
evaluating the star exponential is extremely rudimentary and
at present only star exponentials with general quadratic
arguments of p and x have been calculated,
Maillard (1982).
cf. Bayen and
Also in evaluating the star exponential we
had to solve a partial differential equation which is
somewhat contrary to the spirit of the path,integral.
19
There are two other techniques proposed for evaluating
path integrals.
One is Lee's
(1976) continuum calculus.
It
is deficient in that the endpoint dependence must be
extracted somehow and it still requires the evaluation of a
continuous determinant.
The other method which has recieved a lot of attention
is the Fourier transform technique of Dewitt et aI . (1979)
and Mizrahi (1976).
At the present time this method may-
only be used to calculate Gaussian path integrals.
Its
virtue is that it supplies a somewhat rigorous definition of
the path integral without a time-slicing procedure.
20
CHAPTER 3
THE METHOD OF DELTA FUNCTIONALS
At this point we have surveyed the present techniques
of path integration.
We now wish to present a new technique
called the method of delta functionals and use it to solve
the general Schroedinger equation (E q . (1.2)) where H has
the specific form
(3.1)
I
id tK (q, t I a ,r ) = [F (p , t) - - (qG (p , t)+G (p , t)q) ]K (q, t I a ,r )
This seems to be a somewhat simple partial differential
equation but we will show how when specialized to second
order in p the solutions to this equation can acc ommoda te
nearly all of the known propagators including the following
potentials.
V (q)
-- - qF (t )
The forced harmonic oscillator.
V(q)
I? +
s:
The inverse quadratic potential.
2
V(q)
?
V(q)
q2
e_
I
el
The Coulomb potential.
+
The Morse potential.
Hence as a byproduct of our investigation we will bring
to light the underlying equivalence of all of the propagator
solutions to the above potentials.
Specifically;
in the
subsequent sections we concern ourselves with finding the
21
general propagator solution to E q . (3.1).
To achieve this
end we will write the solution as a path integral.
To
evaluate the path integral we develop a new technique,
the
method of delta functionals (MDF), and apply it to our
problem.
In the next chapter with the solution in hand we will
construct three canonical transformations that allow us
generate further solutions to more complex Hamiltonians than
that of E q . (3.1).
With the presentation of each transfor­
mation we work a physical example
(where H is 0 (p^)) illus­
trating the use of the formalism.
The idea of MDF was conceived by us after examination
of a calculation by Katz (1965).
The basic observation
there was that phase space path integrals with Hamiltonians
that only depended upon momentum (H = H (p)) were easy to
integrate because the functional
integration over position
gave rise to an infinite product of delta functions. In MDF
we extend this idea to Hamiltonians that depend linearly
upon position and we present our reasoning without the use
of time-slicing.
To begin we write the phase space path integral repre­
sentation for the propagator solution to E q . (3.1)
(3.2)
K(b,sla,r) = /[dqdp/2ji] exp(i/* pq - F (p) + qG(p) dt) .
Notice that F and G may be arbitrary functions of time but
we shall rarely make this explicit.
22
If we perform an integration by parts in the exponent
we may write K as
K = /[dqdp/2jr] exp(ipql exp(i/ q(G - p) - F dt)
(3.3)
where we use the notation p q I = p(s)b - p(t)a.
The functional integration over position is a represen­
tation of a delta functional.
Recognizing this allows us to
rewrite K as
K = /[dp]/2% exp(ipq| &[p-G(p)] exp(-i/ F dt )
(3.4)
To proceed any further requires an interpretation of the
delta functional.
Heuristically its presence forces the
contributing paths in momentum space to obey the equation of
evolution
-- = G (p ,t).
(3.5)
dt
The solutions to this first order differential equation are
trajectories that may be labelled by the initial condition
Pq .
In the spirit of a sum over paths we may expect to be
able to replace the momentum functional integration by an
equivalent integration over pQ which labels all of the
contributing paths.
Schematically we have
/[dp] 5 [p-G(p)] f[p] — > Zdp0 f[p(p0 )]
(3.6)
where f is some arbitrary functional and p(p ,t) is the
solution
to p = G(p,t) with initial condition pQ .
However
there is a correction to the naive replacement scheme of E q .
(3.6).
The propagator must satisfy the semigroup property
Eq. (1.5).
Inserting our replacements for E q . (3.4) into
E q . (1.5) gives
23
Zdq0/2 jt db dpQ/2n exp(i(q(t)c-q(s)b) exp ( i (p ( s )b-p ( r) a )
x exp(-i/ F(q(q0 )) dt) exp(-i/ F(p(p0 )) dt). (3.7)
The b integral is a representation for a delta function so
we may write E q . (3.7) as
/dq0/2jr dp0 e xp (i (q (t )c-p (r)a ) exp(-i/dt F)exp(-i/dt F)
x 8 (q(s) - p(s)).
(3.8)
Writing the delta function in terms of qQ
8 (q(s) - P (s ) ) = M q 0 - pD ) (dq (qQ, s )/ dqQ )-1
(3.9)
and integrating over qG gives
/dp0/2n (dp(pG )Zdp0 )-1 exp(ipq|
exp(-i/ F(p) dt).
(3.10)
Clearly to preserve the semigroup property we must include a
factor (dp(p0 ,s)/dp0 dp(p0 ,r )/dp0 )
^2 in our replacement
scheme to cancel the factor arising from the delta function.
Hence the correct replacement is
dp(p0 ,s) dp(pQ ,r)
/[dp] 8 [p-G(p)]
f[p] — > Zdp0
dP0
x f[p(pn»t )]
dPo
(3.11)
Using this result in E q . (3.4) we may write the general
solution to E q . (3.1) as
K (b . s i a . „
- /dp /2»
(3 .12)
x exp(i(p(pc ,s )b-p(p0,r )a ) exp(-i/r F(p(p0,t) dt)
where p(p0 ,t) is the solution to the equation of evolution
dp
G(p,t)
(3.13)
dt
with initial condition p0 = p(t = 0 ).
One might feel a lack of confidence in our solution due
to the heuristic reasoning involved.
However it is easily
24
verified that this is the correct solution by simply chec­
king to see that it does indeed satisfy E q . (3.1) .
In
Appendix A we present a more rigorous derivation of the
replacement scheme E q 0 (3.11) using a time— slicing defini­
tion of the path integral.
To illustrate the method we present the example of the
free particle.
The propagator is written as a path integral
K (b la) = / [dqdp/ 2 Ti] e xp (i f q
pq - p^/2 dt) . .
(3.14)
Integrating by parts and recognizing the delta
functional allows us to write
K = /[dp]/27t e xp (ipq I e xp (- i/ p^/2 dt) 5 [p ] .
The solution to dp/dt = 0
(3.15)
is p = constant = p 0 and the
semigroup factor dp(pG )/dp0 = I.
Hence we may write K as
K = / dp0/2jt exp ( i (pQ (b-a) - / q p 02/ 2 dt).
(3.16)
Performing the time integration in the exponent gives
K = / dp0/2ji exp( H p 0 (b-a) - p 02T/2)
(3.17)
and integrating over p0 leads to the. result
K(bla) = (27tiT)“1/2 exp (~ (b-a)2 ) .
(3.18)
We now make some remarks about the technique.
easily generalized to higher dimensions.
It is
In doing so, one
will encounter matrix equations of evolution with solutions
of the form
p(t) = A (t ) .p0 .
(3.19)
The semigroup factors in this case are easily found to be
[detA(s)detA(r)]l/2 .
25
Also as we shall see one often deals with delta
functions of complex quantities.
simply ignore this fact.
The prescription is to
The integration is always over
real p @ .
As noted,
tion by Katz.
this technique was suggested hy an observa­
In our notation his observation (based upon
time-slicing) was
/[dp] 8 [p] F[p] = / dp0 F[p0 ],
(3.20)
Recently we uncovered one other paper that contains formulae
of this sort.
This is the article by Cambell e t a I . (1976) .
Using time-slicing they derive the formula (in our notation)
/ [dp] 5 [p-G(p)] BiPxl
(3.21)
IT
9G
= / dp exp{i (pb-p(0)a )}exp{- -/ ds Re — (p(s),s)}
2 0
9p
where p(t)
is a solution to dp/dt = G(p).
If one compares
this equation with E q . (A.36), one sees that this is
equivalent to our formula (restricted to F = 0) except for
the requirement that the real part of G'(p) be used.
is incorrect.
We also feel it is simpler to write the
semigroup factor as (dpZdpg)^/^.
This
26
CHAPTER 4
CANONICAL TRANSFORMATIONS
Now that we have a new method for calculating path
integrals we wish to extend our technique so that we may
move beyond the restriction of Hamiltonians linear in
position.
To this end we will introduce in this chapter
three canonical transformations and illustrate each with a
worked example.
In all cases our strategy is to transform
the Schroe dinger equation and then write down the path
integral that corresponds to it.
I
This may be done because
our time— slicing definition of the path integral enables us
to determine the Schroedinger equation to which the path
integral is a solution.
We refer the reader to the discus­
sion concerning E q . (1.18).
Consequently we suffer from no
factor-ordering ambiguity in our derivations.
We shall see
that in general when one transforms the path integral,
correction terms must be added to the naively transformed
'
I
I
Hamiltonian.
In most cases this correction may be cast into
the form of an effective potential.
LINEAR MOMENTUM TRANSFORMATION
The first canonical transformation that may be
applied to our phase-space path integral in order to bring
it into a more convenient form is a linear transformation
27
in p (t) .
p = P +
f (q) , q = Q, dp = dP, dq = dQ .
K (b |a) = /[dq dp/2n] exp(i / pq-H(p,q)
(4.1)
dt)
(4.2)
= /[dQ dP/ 2 Ti] e xp (i / (P+f(Q))& - H(P+f(Q),Q)
In general for Hamiltonians of higher order than p
dt).
a
correction term U must be added to H to preserve the correct
)
ordering of the quantum operators.
We do not calculate U
for this transformation as all physical examples are O(p^).
Correction terms must be calculated for the transformations
we discuss later in the chapter and it will become apparent
from their evaluation how the correction for this transfor­
mation would be calculated.
As shown in Appendix A we have the useful results:
/® f (q)q dt = /g f (q) dq = gI
(4.3)
where dg/dq = g'(q) = f(q) and gI is defined to be gI =
g(b)-g(a).
Consequently such a term will be a constant as
far as position integration is concerned.
We also have an
integration by parts identity
/ S pq dt = p q I - / S pq dt
r
r
and note that [dq][dp/2n] = [dq/2 it ] [dp ]/ 2 jt .
(4.4)
To illustrate the utility of this transformation we
work out the propagator for the standard one-dimensional
harmonic oscillator.
S = /
y
The action is
.
p^
M^q^
pq - -— - — —- d t .
2
2
By performing a linear transformation in p (basically
completing the square)
(4.5)
28
p = P -
iwQ, q = Q,
(4.6)
w e may write the action in the more convenient form for use
in the path integral
S = / Pd - i(i)Qd - P2 / 2 + im PQ dt.
(4.7)
Inserting this into the path integral and integrating
out the total
derivative gives
(4.8)
2
K = exp (
I )/[dPdQ/2xr] exp(i/ Pd - P2 /2 + iwPQ dt)
At this point we note that this path integral is still
a solution to the appropriate Schroedinger equation.
Eq. (1.25) K i s a
a
i— K
at
=
From
solution to
W
I
I
I
I
+
a
a
a
[(— i--HtoQ)
2— (Qid) (— i-- Hi(i)Q) + (— i-- H itoQ) itoQ) / 2 ]K
BQ
BQ
8Q
i a2
W2Q2
(4.9)
2
2 BQ2
Using M D F , E q . (4.8) may be immediately rewritten as
2
K = exp ( " - I ) / dp0/2TT eiPQl (Ip Iil ip l£l)i/2
dPo
dPo
(4.10)
x exp(-i/® P 2/2 dt)
where dP/dt = iwP.
The solution to this differential
equation is P(Pn,t)
P ne
o'" = ~o
.
•
Choosing6 tn
O = (s + r)/2
causes the semigroup factors to equal unity
(
(4.11)
1/2 = ^e im (s-r) / 2e id) (r-s )/2)I/2 _ 1
dPo
dPo
The time integral in the exponent is easily done giving
-i/® P2 /2 dt = - — o sin(&)T)
1
2d)
with T = s — r. Hence
(4.12)
(4.13)
K(b,s|a,r) = exp(^ (b2-a2 )) / dP /2n
x exp(iP (be16) ^ -ae iwT/2)) exp(---o sin(o)T))
2d)
to
.1/0
. id)
)^/2 exp(
= (
((b2 + a2 )cos(toT)-2 ab))
2n isin(m T )
2 s in (d)T)
29
which is the usual expression for the harmonic oscillator
propagator
(see Eq.
(2.15).
To use this transformation,
must be clever in completing the square.
one
This is done for
various propagators of physical interest in Appendix C .
POINT CANONICAL RESCALING
The next canonical transformation we examine is a point
rescaling of the position.
Let
(4.14)
p = P / f ' (Q) , q = f (Q) , dp = dP/ f ' (Q) , dq = f ' (Q) .
2
For convenience we also define g(Q) = f'(Q) . We interpret
g (Q) as a kind of 'metric' on a one dimensional
space.
Then
K (b la) = /[dq dp/2n ] exp(i / pq - H (p ,q) dt)
(4.15)
is transformed to (b = f (B) and a = f (A ) )
K(BlA) = (g(B)g (A) )-1/4 /[dQ dP/2jt] exp(i / P& dt)
x exp(-i f H(g(Q)_1/2P ,f(Q)) + U(P,Q) d t ) .
(4.16)
II(P1Q) is an correction term that must be included for the
following reason.
The rescaled K is a solution to the
rescaled Schroe dinger equation where the Hamiltonian (H) is
Hg(g-*/4Pg-*^4 ,f (Q)).
However
our path integral
is a
solution where the Hamiltonian has its operators ordered
according to S.
We define a symmetrizing operation Sym ( )
which takes the operators in a function and permutes them
until the symmetrized form is reached,
Eg (p *q) such that hg(p, q) = Sym( f (p, q)).
nian used in our path integral must be
ie: there is some
Hence
the Hamilto­
30
(4.17)
Sym(Hs (g-1/4Pg-1/4 ,f(Q)))
and this is in general not equal to ITg(P1Q) where
(4.18)
H(P,Q) = H (g 1/2P,Q)
However, one may write
S S(P,Q) + U(P.Q) = Sym(Hs(g~1/4Pg 1/4 ,f(Q))
(4.19)
and interpret U as a necessary quantum correction needed to
maintain a specific operator
ordering.
As an example we wor k out the case for H = p 2/2m +
V (q).
Then
t> -1/4 ,f) = (2m) 1 (g 1/4Pg 1/2Pg 1/4 )+V. (4.20)
Hg (g-I/4
"*■' nPg
Using the commutation relation [P,Q]
— i one finds
Sym(Hg(g-Pg-)) = (2m)-l(g-lp2+P2g-l)/2
_ _§— , +
— + Ye
8mg2
32mg ^
Now -H(P1Q)
P2/ 2mg + V SO-Hs(P1Q) = (2 m)
(4.21)
(S-1P2+?2 g-1)/ 2 +
V.
It is now easily seen that the necessary U is
,2
U(P1Q) = ---— (——— — g '').
(4.22)
8mg2 4g
The example we work to illustrate this transformation has an
inverse quadratic potential.
Examine the action
(4.23)
2
, m .o m o o
„o
r
o p
m o o
-j
-g / yZ dt = / py - — - - Q jly z -g/y 2 dt
S = Jf -v
-- -Qzyz
-SI j
-g/y
2
2
2m
2
where Q i s a function of time £I(t). To place this Hamilto­
nian into an appropriate form for the use of MDF we perform
a rescaling point canonical transformation.
(1+aq)1^ 2p,
is (m = I )
Let p ->
y — > 2(l+aq)1^2/a. Then the rescaled action
31
S = /
pq - (1+aq)p^/2 - u^q^/2(1+aq) + qF - w^/a^ + F/a
+ a2/32(1+aq) dt
(4.24)
where e = 2 ft(t ), e 2 = w 2 - 2aF (t ), and g = 2to2 /
Note that m and g are constants.
- 1/8.
Expressed this way the
particle appears to be a forced harmonic oscillator with a
position-dependent mass.
The subtraction of the effective potential term
U = a2/32(l+aq) neatly cancels the last term of the action.
As it stands we must still mod i f y the action if we are
to apply MDF,
so we perform a linear transformation letting
p - > p + iroq / (1 + aq) which allows us to write S as
S =
iM(b-a)/a - (im/a2)In{(1+ab)/(1+aa)} - M 2T/ a2
+ /® pq - (l + aq)p2 /2 - i M p q + qF + F / a dt
(4.25)
and the path integral we wish to evaluate as
K (b Ia ) = (---b-)<’)/(^ (l + ab)1 / 4 (l + aa)1/4
1+aa
x e xp (-oi (b - a)/ a - Im 2T/a2) e xp (i/ F (t)/a dt)
(4.26)
x / [ ~ — ^-] exp {i/dt pq - (l + aq)p2 /2 - iMpq + F q } .
2 JT
Notice the introduction of the fourth root factors as per
Eq. (4.16).
The pre-integration factors will be called PI.
Integrating by parts in the exponent and noting the delta
functional
gives
(4.27)
2
K = (Pl/2 jt )/ [dp ] 8 [p + )3p2 + iMp-F] e
^ e~"
p ^2
where p = a/2 .
The solution to dp/dt = F (t) — iMp — Pp2 may be found
through a dependent variable substitution.
Define a
32
function f(t) by
p(t) = i(t)/pf(t) - ito/2p.
(4.28)
Substitution of this form for p(t) into the differential
equation for p leads to an equation for f (D^ = d^/dt^)
D2 f (t) + (to2/ 4 - pF(t))f(t)
= 0.
(4.29)
This is the equation for a harmonic oscillator with a time
dependent frequency e/2 = £2.>
contained in Appendix D.
A discussion of this system is
To solve this equation (formally)
we construct a Green's function which satisfies
(D2 + fi2)G(t,s) = 5(t-s),
(4.30)
then the solution f(t) may be expressed in terms of f at an
earlier time s by
f(t) = G( t , s )TTsf (s )
where AUtB = AdB/dt - (dA/dt)B.
(4.31)
We are dealing with the
retarded Green's function G(t,s) = 0 if t < s.
may
write
Using G we
the solution p(pQ,t) as (Gts = G(t,s))
I Gts(j}po + ito/2) - Gts
ito
(4.32)
with the 'dot' signifying time differentiation.
The semigroup factors are found by differentiating the
above equation for p with respect to the initial condition
P0«
The expression may be shown to be
d p (t )/dpn = (fn/f(t))2
(Gto(pp0 + ito/2) - Gto) 2
(4.33)
where we have used the constancy of the regular Wronskian to
set (see E q . (D.12))
GtsGts — GtsGts = I.
(4.34)
33
Using MDF and taking the arbitrary initial time t 0 = r
allows us to rewrite the path integral as (p = p0)
K = (Pl/ 2 jt) /dp
(f r/fs) eipql e xp {- i/ p2/2 dt}.
(4.35)
The time integral in the exponent may be shown to be
S
-ip F (t )/ a dt + i(p(s)-p)/a - (2a)/a2 ) In (f g / f r )
+ iw2T/a2 .
(4.36)
By defining a new variable of integration u = f g / f r and
using the above result for the time integral we may after
some algebra put E q . (4.35)
into the form
2
K = (2jrpGsr )-l ( (1+ab) (1+aa) )*/4
M /a
1+aa
x, e xp {(2 i/C2Gsr )((1+ab )Gsr - (1+ aa )Gsr ) }
x f du u~ (U+I ) exp{(-i/PGsr)(Au + B/u )}
with B = b + 1/a etc.
The remaining integral
(4.37)
is a represen­
tation of a modified Bessel function of order |i = 2w/a2 and
recognizing this allows us to write K as
K = (2/iaGsr )((1+ab )(1+aa ))■*•/^
x ex p {(2i/a2Gsr )((1+ab)Gsr - (1+aa)Gsr)}
x I jlC (-4i/a2Gsr) ( (1 + ab) (1+aa) )1/2} .
(4.38)
W e may rescale this back to the original position variables
of interest 1+ab — > a b /4 to write finally the propagator
for the inverse quadratic potential with time dependent
frequency as (p = j (I+Sg)
2)
_ K (b » s I a ,r ) = (ba)^^2 (iGsr)-*
i
ib a
x e xp{--— (b2Gsr - a2Gsr)} I {- ---}
2G sr
H
Gsr
(4.39)
which agrees perfectly with the result given by Khandekar
and Lawande
(197 5).
J
34
For constant w = fi, we may use the explicit expression
for Gsr = sin(uT )/m to rewrite K as
K (b , a) = (ba)l/^ (m / isin(toT) ) e xp {i-c o t (w T ) (b ^ + a^ ) }
1 1I-1" 1
(4,40>
There is an interesting limit to this propagator that
We would now like to discuss.
happen in the limit g — > 0.
W e wish to see what would
Naively one might have assumed
that this would reproduce the harmonic oscillator result.
However we shall see that this is not the case.
— > 0 is equivalent to ji — > 1/2.
The limit g
For fi = 1/2 we may -
introduce a simple expression for I11
I1/2 (-iz) = e“ i3jt/4 (2jrz)“ 1/2 (eiz - e“ iz)
(4.41)
and substituting this form into E q . (4.40) gives
K = ((o/2 jti s in (<oT) )•*•/2 {e xp (-- - 7 — - ( (b 2 + a2 )c 0 s (w T )-2b a ) )
2 s in ((oT)
(4.42)
((b2+a2 )cos(m T)+2b a ))}
eXp(2sin(toT)
which may be recognized as
K(n = 1/2) = KH0 (bla) - KH0 (-bla) .
(4.43)
KgQ is the propagator for the harmonic oscillator given by
Eq„
(4.13).
From a discussion presented in Chapter 6 (see
Eq. (6.20)) we note that the form given by Eq. (4.43) is
that of the propagator for a harmonic oscillator with a hard
reflecting wall at the origin.
This might have been antici­
pated from the fact that for non- zero g there is always a
hard core at the origin preventing the particle from travel­
ling from the positive side of the axis to the negative side
and vice-versa.
Hence since for any non— zero value of g
35
there is a hard core, w e must have a hard core in the limit
as g goes to zero.
This fact has been thoroughly discussed
by Klauder (1979) but in a path integral context has given
rise to an erroneous statement by Schulman (1981, p. 345;
that the limit is the HO) and a m i s g u i d e d paper by Langguth
and Inomata
(1979).
These latter authors try to obtain the
naive HO limit by essentially modifying the Bessel function.
This is an incorrect approach.
There is no need to modify
the Bessel function as the naive limit gives in fact the
wrong result.
A hard core must persist.
TIME RESCALING
We now turn to a third transformation that may be used
to evaluate path integrals. This is a rescaling of time
first introduced in connection, with path integrals by Duru
and Kleinert (1979) in their calculation of the propagator
for the Coulomb potential.
however,
Our derivation of this rescaling
differs substantially from theirs.
Examine the path integral form for the propagator:
K = / [dxdp / 2 jr]e xp {i/ px - H dt} . .
(4.44)
This propagator is a solution to (p = -idx)
Hg(p,x)K = idtK.
(4.45)
Let us now insert the unit constraint functional
F [x (t)] = I = /"^ dsb 5(sb - sa - / f(x(t)) dt)
and we remark that (S = sb - sa )
sb
8 (S - / f dt) = 8 (T - /
(f)-1 ds )/f(b) .
sa
This allows us to rewrite Eq. (4.44) as
(4.46)
(4.47)
36
K = / ClSjj f (b
[dxdp/2jt] 8 (T - f ds/f)
x ex p {i/ px - H dt}.
(4.4 8)
The presence of the constraint enables ns to use a new time
variable s with ds = f dt.
We write
K = /00 ds. f(b)-1 [dxdp / 2it] 8 (T - / ds/f)
sa
x exp{i/ px - H/f ds}
(4.49)
where we have used the 0 (T) = (HS) to rewrite the lower
limit of the ds^ integration.
Now we insert the, integral
representation of the delta function to write
K = /” dS f(b)“1 /" dE/ 2« eiET [dxdp/2jt]
0
— CD
S
x ex p {i/^ px - H/f - E/f ds}.
(4.50)
We have changed variables from s^ to S in the above.
The asymmetry implied by the f(b) term poses the
question whether K as written still satisfies the
Schroe dinger
equation Eq.
(4.45).
The answer is no.
Duru and Kleinert get around this by using an ad hoc
averaging procedure to eliminate the asymmetry.
Pak and
Sokmen (1983) also use an average to write a symmetric
integral.
The correct way to eliminate the asymmetry is to
make sure K that continues to satisfy Eq.
(4.45) which
implies the addition of correction terms to the Hamiltonian
just as we did had to do wi t h the point canonical rescaling
transformation.
The correction term U can be found from
requiring the quantum Ug to satisfy
Ug = H( f)—1 - S y m ( H U ) - 1)
which ensures that Eq.
(4.4 5) is satisfied.
(4.51)
For H(p,x) of
37
the form
H = pZ /2g(x) + V(x)
(4.52)
we find (remember U <, is a quantum operator and ' = d/dx)
;2\—1 + (g f,2
f'
,p ) + -g 'f '(g f)-2^
Uo = -(pf'(gf^)
^ \-I f
(4.53)
°
4
4
with corresponding classical correction U to the path
integral Hamiltonian
ipf '
I g 'f '
U ------ + — ----- .
(4.54)
2g f2
4 g2 f2
It is of interest in the case where H is of the form
Eq. (4.5 2) (and we will restrict ourselves to this case from
now on) to eliminate the cross terms arising from the corre­
ction U.
This process will also result in the explicit
elimination of the apparent asymmetry (which has been dealt
with through the addition of U but is not manifest).
To
cancel the cross terms we make a linear transformation and
let p — > p - if'/2 f.
vatives
gives
Integrating out the total time deri­
( C = (g(b)g(a))-^"^4)
K = C / dS (f(bjf (a))“ 1/2 dE / 2 n e iET [dxdp/2jr]
(4.55)
,2
§.L1L _ YI e A _ Y ds } .
x ex P {i/ px ~
f
f
2gf
Sgf3
4g2f2
Notice that the correction terms now take the form of an
effective potential and that the symmetry in the endpoints b
and a is now manifest.
Suppose the f was chosen to cancel the g in the p term.
This would lead to a propagator K
K = C-1 f dS dE/2it e iET [dxdp/2n]
x e xi
cpfi/ px - p2/ 2 + g '2ZSg2 - gV - gE ds}. (4.56)
38
The effective potential is now simply g'^/8g^„
Also note
that the endpoint factor is now C
At this point it would be instructive to wo r k an
example.
Let V(x) = e/x.
The path integral we wish to
evaluate is
K = /[dxdp/2jr] exp{i/ px - p^/2 + e/x dt}.
Let ds = dt/x.
The correction U = -ip/2.
(4.57)
Substituting this
into the above and rewriting in terms of s and S gives
K = // dS b dE/2jr e~iET [dxdp/2n]
x expti/ px - xp^/2 + ip/2 + e + Ex ds}.
(4.58)
We turn to the evaluation of the interior propagator G
G = /[dxdp/2jr] expti/ px - Hx - U + xE ds}.
(4.59)
Using MDF we may rewrite G as (neglecting factor e*e^)
G = /[dp]/2% eiP=l S[p + p2/2 - E] e~^ P /2 d s .
(4.60)
This path integral is very similar to the one w o r k e d in the
evaluation of the inverse quadratic propagator (see for
example Eq. (4.27)) and we
calculation.
omit presenting a detailed
Using the methods illustrated by the previous
example we may solve the differential equation in the delta
functional,
find the semigroup factors,
integration in the exponent with result
and perform the time
(including e*e^)
G = e*e® exp Iiacth((I)S) (b+a) } (2na/isinh(wS))
2 ia(ba)d/2
1{
sXnh"(ws7~} ‘
(a/b)^-^2
(4.61)
We have defined a = (2E)d^2 and <o = (E/2)"*"^2 .
Substituting this result into Eq.
e xpre ssion
(4.58) gives
the
39
(isinh(toS)) ^
x exp{iacth(toS)(b+a)} I^{-2ia
(4.62)
s inh(MS)
for the V = e/x propagator.
We point out that this time transformation enables us
to write the propagator of interest as the double Fourier
transform of another propagator which may be easier to
evaluate.
We see that we might profitably combine the point
rescaling transformation with the time transformation to put
the previous statement into practice.
Duru (1983) has used
this idea to evaluate the propagator for the Morse potential
and this concept is the subject of the w o r k of Pak and
Sokmen.
Examine the propagator
K(b|a) = /[dxdp/2jr] exp{i/ px - p^/2 - V(x) dt}
when rescaled by x = f(q) (g(q) = f'^).
(4.63)
With b = f(B) and
a = f (A) we may reexpress K in terms of the rescaled q u a n t i ­
ties a s
K(BlA) = (g(B)g(A)) 1/4 /[dqdp/ 2jt]
x expti/ pq - p^/2g - V(f(q)) - V e dt}
(4.64)
where the effective potential V e is
V e = 9 g ,2/32g3 - g " / S g 2 .
(4.65)
Now we perform a time rescaling to eliminate the g from the
kinetic term in the Hamiltonian.
Let ds = dt/g,
then (4.66)
K(BlA) = (g(B)g (A) )1/4 f f dS dE/ 2 jr eiET [dqdp/2%]
x exp{i/ pq - p2 /2 - V(f(q))g(q) - Eg(q) - V g }
with a new effective potential V g given by
40
V e = 5g'2 /32g2 - g " / 8g.
(4.67)
This result has also been recently obtained by Pak and
Sokmen through a different method.
They find the correction
term by examining variable transformations in the timesliced path integral.
This is a notoriously subtle calcula­
tion (see for example the three differing values of the
correction term given by McLaughlin and Schulman (1971),
Gervais and Jevicki (1976),
and Gerry (19 83) for the same
time— sliced path integral) and is only applicable to Hamil­
tonians quadratic in momenta.
Pak and Sokmen use the result
of Gervais and Jevicki (which appears to be the correct one)
to obtain precisely the correction given by Eq. (4.66).
Our
method gives an independent check of their work while provi­
ding the physical m o t i v a t i o n for the need of such a cor r e c ­
tion term.
It is also more general in that it is again not
restricted to quadratic Hamiltonians.
Also it is an impor­
tant check of the consistency of this transformation since
we use a quite different time— slicing definition of the path
integral than they do.
There can be no factor— ordering
ambiguity in the expression for the effective potential so
it is significant that both methods lead to the same result.
41
CHAPTER 5
RELATIVISTIC PROPAGATORS
We now extend the formalism of MDF to relativistic
propagators.
There are two approaches one may take.
We
briefly describe one technique and then turn to the second.
The first method merely uses the relativistic action in
Hamiltonian form.
However this is not straightforward.
The
relativistic action S in Lagrangi an form is
S =
m(x^)^/^ ds .
The variable s is any path parameter.
time is used which requires s such that
(5.1)
Sometimes the proper
= I.
The
canonical momentum is p = m x ( x ^ ) ^ ^ with consequence
p^ =
„
The Hamiltonian is p.x — L = L — L = 0!
This
occurs because the Lagrangi an is invariant under arbitrary
time rescalings s — > t(s) which requires an identically
vanishing H . - Hence the action in Hamiltonian form is
naively
S = -/ p.x ds,
but this cannot be right because the condition p
(5.2)
= m
is
not enforced and does not arise from the equations of
motion.
This may be corrected through the introduction of a
Lagrange multiplier a and writing S as
S
-/ p.x
a (p2 - m2 ) ds
(5.3)
42
The multiplier o is to be treated as a new independent
variable.
The path integral corresponding to this action
for a free relativistic particle is
K = / [dxdp/2jt] [da] e x p { - i f p.x - a (p^ - m^ ) ds } . (5.4)
The multiplier is functionally integrated over giving rise
to a delta functional 8 [p^ - m^] ensuring that only paths
that satisfy this relation contribute to the path integral.
While this method is probably the correct way to define
relativistic path integrals there is another, more conve­
nient technique that uses the fifth parameter formalism of
Schwinger (1951) and Feynman (1951).
That is the technique
we now describe and use in our example. The demonstration of
the equivalence of these two techniques using the FaddeevPopov determinant is shown in the papers of Bardakci and
Samuel (1978) and Krausz (1981).
To begin we examine the Kl ein-Gordon equation that
a relativistic propagator must satisfy,
+ m^)K(x|a) = 8 (x-a).
(5.5)
In operator notation with G = 9^ + m^ this takes the form
GK = I .
(5.6)
He n c e
K = G " 1 = i/“ ds e" isG .
(5.7)
Taking matrix elements of both sides using < b I and Ia > gives
<bIK Ia > = K (bla) = i f ds < b Ie~l sGIa>
(5.8)
and recognizing the term on the right-hand side as simply
the Schroe dinger representation of a propagator with
(5.9)
Hamiltonian G allows us to write K in the form
K (b la) = i f q ds f
[dxdp/2jr] exp { - i f ^ p.x - p^ + m^ dt} .
We may apply the method of MDF to either of these forms
(E q .
(5.4) and E q . (5.9)) for the relativistic propagator but the
second equation usually turns Out to be the more convenient.
As an example we work out the propagator for a particle
in a constant electromagnetic
field tensor F = constant.
(EM) field where the Maxwell
The action S is (notice that t
is not time, merely a path parameter)
S = -'/ p . x —
(p - eA) ^ + m^ dt.
For a constant F , A =
-F.x/2.
(5.10)
To place this expression into
a form in which MDF may be used we must cancel out the
quadratic expression in A ^ .
To do this we remark upon
several identities for F = constant.
First we define the
dual of F , F* = gOpP^Fp^/2 and the complex vector n in terms
of the constant E and B fields, n = E + iB = n4i.
Then
FT = -F and F*1" = -F*,
(5.11)
FF* = F *F = E.B,
(5.12)
F2 - F *2 = E2 - B2 ,
(5.13)
(F + iF*)2 = (E + iB)2 = n2 = n2 ,
(5.14)
exp{a(F+iF*)} = cosh(an) + (F+iF*)sinh(an)/n.
(5.15)
perform a linear transformation letting
p — > p — — F(F+iF ) .x
2n
which transforms the action into
(5.16)
44
S = -f
dt p .x - — x .F (F+ iF*).x + — x.F^.x
2
2n
4
----- x.F(F+iF*)2F.x - ep.F.x + -p .F (F+iF’’) .x
4 Dl2
2
n
----x.F(F+iF )F.x + m2
(5.17)
4n
= — x .F (F+iF*).x I - m2 s - / p.x - p2 - ep.N.x dt.
4n
where we have defined the matrix N as
N = F (I - F(F+iF*)/n).
(5.18)
Using MDF we may reduce the interior path integral in E q .
(5.9) to quadratures
2
.
i
exp { (i e/4n) (x .F (F+iF*) .x I} e im s (2 jt) * I d^pQ e
*x *
x ex p {i/ p2 dt}
(5.19)
with p (t ) satisfying p = - ep.N.
The succeeding steps are the standard ones of solving the
differential equation in terms of p Q, calculating the
semigroup factors
(which equal I if the symmetric time
limits s/2 and — s/2 are used for the time integral in the
exponent), evaluating the time integral in the exponent
(Eq.
(5.15) is useful), and then performing the integral over the
initial momenta using Eq. (B .5) . The result for K is
2
K = i/^ ds e~ ^in s (4jt)-2 de t [ ieF/s inh (e sF)]-^ 2
x exp{(-ie/4)(b-a) .Fcotanh(esF).(b-a)} .
(5.20)
45
CHAPTER 6
GROUP MANIFOLDS
In the previous chapters we have discussed the method
of delta functionals and canonical transformations for nonrelativistic and relativistic path integrals.
In all cases
the underlying manifold was the real number line.
In this
chapter we discuss what happens when the manifold is changed
to a circle or the half line.
We do this to illustrate how
MDF may be generalized to cases when the momentum is dis­
crete or doesn't exist.
Our examination will also provide a
deeper understanding of what is involved in the construction
of a path integral.
We first examine the case of the
circle.
The manifold of the circle is the interval
the endpoints identified.
(0,2jr) with
This creates a compact manifold
which as a consequence requires the momentum to take on
discrete values.
Marinov and Terentyev (1979) suggest a way
to deal with this complication.
Their prescription is to
extend the range of position integration from (0,2jt) to
(-00,00), to work out the corresponding propagator,
and then
to take that propagator and sum over all homotopicalIy
equivalent classes.
we have
For example in the case of the circle
46
K = / 7tEdti] exp{iS}.
(6.1)
O
Extending the limits of integration creates a new propagator
G = /_ [dG] exptiS}.
(6.2)
To find K in terms of G(A G1T) where AG is in (-<»,'<») we
define a new variable AG = Aa + 2 n n where Aa is in (0,2%)
and n = +1, +2, etc.
The hom.otopical Iy equivalent classes
are labelled by n (the winding number) so we have
K(Aa1T) =
2 ”ooG(Aa + In h 1T).
(6.3)
A word about 'homotopicaIIy equivalent classes'.
We might
as well have said 'quantum mechanically equivalent classes'.
An observer is blind to any revolutions the particle might
have made around the circle.
and final endpoints.
One only 'sees' the initial
Hence a trip from a to |i without a
revolution and one from a to p with n revolutions are quan­
tum mechanically equivalent.
The dictum
of quantum mecha­
nics is to add up the amplitudes for all possible ways an
event may occur,
in this case to sum over all revolutions or
'homotopicalIy equivalent classes'.
We may formalize this prescription.
Let H be a Hamil­
tonian which takes its values on the manifold of some group
R (usually the real number line).
Suppose H is invariant
under the action of a group G with elements g and let M be
the manifold
(coset space)
of the factor group R/G.
Then we
may write the propagator of the manifold M in terms of the
propagator of the manifold R by
47
KM ( b | a ) = 2 K ] j ( g b l a ) o
(6.4)
In the case of the circle, R is the real number line
which is the group manifold for the non-compact translation
group T with elements e ^ .
Translations by 2 Rn form a
subgroup Z and the factor group T/Z is the group U(I) with
elements e*®..
circle
(0,2n).
The group manifold for U(I)
is just the
The action of a group element g of Z upon b
is just gb = b + 2nn and so
KM (bla) =
2 KR (gb!a) =
2 %
Kr (b + 2nn|a).
(6.5)
We may generalize the construction of Marinov by adding
in a phase factor c so that Eq. (6.4) becomes
Km ( b I a ) =
2 c(g)KR (gbla) .
(6 .6 )
The c (g ) are essentially determined by the boundary condi­
tions that Km must satisfy.
We now wish to construct the path integral on the
circle and see that it does reduce to Eq. (6.5).
so we will generalize MDF.
In doing
To begin we must redo the
derivation of the path integral illustrated by Eq. (1.13).
The propagator may be written as
K(FiIa) = (II /Q ,rden )<nl6N_ 1 >...<U1 la>.
(6.7)
We need to evaluate the infinitesmal propagator
which in the Schroedinger representation is
<en + l le" iHAt|en>
(6 .8 )
We insert a complete set of momentum eigenstates,
<e|p> = (27T)™1/2 eine
where we have defined n = p which is some integer,
(6.9)
into
48
E q . (6.8)
to give
<6n+2 l6 n > =
2 exp{ipA6 - iH(p,0)At} .
(6.10)
This allows us to write K symbolically as
K(p I a) = / 71
0
(6.11)
2 [d0 ][p/2jr] exp {i/ p6 - H( p ,0 ) dt } .
We illustrate the use of this symbolism by working out
the propagator for the free particle confined to a circle.
For a free particle we have
K=
2 [p/2jt] f
[dO] e xp {i/ p6 - p^/21 dt}
where I is the moment of inertia.
(6.12)
Using the idea of MDF we
integrate by parts and recognize the position integral as a
delta functional to write K as
K =
2
[p]/27t BiP0 I 8 [Ap ] e x p { - i f p2/2I dt} .
(6.13)
The solution to the discrete equation Ap = 0 is p (t ) = p
where p is a constant (remember that p at any time is an
integer).
The semigroup factors are unity.
Hence the
propagator is (writing p as m to emphasize the discrete
nature of p)
K=
2 (2jr)-1 exp {im( p-o) - im2T/21} .
(6.14)
By using the theta function (not the step function) defined
by
6 (a ,T) =
2 T 00 exp {ima - m2T/ 2}
(6.15)
we may write K as
K (|iI a ) = (2jt)_1 © (p—a ,T/1 ) .
(6.16)
The Marinov prescription would have us evaluate the
infinite limit propagator G which is
G = (I/2niT)l/2 exp{iI(fi-a)2/2T}
(6.17)
49
and write K as
K=
(I/2niT)1/2
2 exp {i I (fi+2 nm-a)2/2T} .
(6.18)
The result is actually equal to the expression given by E q .
(6.14) since the theta function satisfies the identity
6 (a „T) = (2 jt/ iT)
2
2 exp {i (a+2 nn )2Z 2T} .
(6.19)
We now turn to a more involved case, that of a particle
confined to the positive real axis.
The real number line R
(excluding zero) is the group manifold for multiplication of
the reals.
A subgroup is
elements I and -I.
the group consisting of the two
Zj is essentially the parity group.
The
factor group M = R/Zj is the group of multiplication for the
positive reals and the manifold is just the positive real
axis.
The generalized Marinov construction gives for the
propagator
KM (b la) = K r (bI a ) - KR (-bla)
(6.20)
where we have chosen the phase factors c (g ) to equal g.
This was decided by the requirement that K vanish at the
wall at x = 0 .
We now wish to derive this result from a construction
of the propagator.
As before, the construction hinges upon
the evaluation of the infinitesmal propagator
<x+ |e"iHAtlx>.
(6 .21 )
However in this case we cannot insert momentum eigenstates
since they do not exist.
This is related to the fact that
there must always be a wave reflected from the wall which
prevents the eigenfunctions from satisfying the boundary
50
condition at the wall where they must vanish.
however construct eigenfunctions
We can,
Iq> of the kinetic energy.
These are
<xlq> = (2/ tt)"^^ sin(qx).
(6 .22 )
They are orthonormal and complete on the positive real axis.
We may insert them into E q . (6.21) to write the infinitesmal
propagator as
/
dq (2/ jr) s in (qb )s in (qa ) g-iHAt
Unfortunately,
(6.23)
this expression may not be concatenated.
To
avoid this we create a special notation and define
Sin[px] = II sin(px+ )s in(px).
(6.24)
This functional admits an integration by parts identity
Sin[px] = sin(p(s)b) Sintpx] sin(p(r)a)
(6.25)
and provides a representation of a delta functional
/
0
[2dp/7t ] S i n [ p x ] = 8 [x]
(for x > 0) .
(6.26)
Using this notation we may write the propagator on the half
line as
K(bia) = /
[2dp dx/Jt]S in [px] exp {-i/ H (p ,x ) dt } .
We now derive the Marinov expression.
(6.27)
Examine the
double integral
/
dx 2dq/jt s in (px )s in (qx)s in (qa) g-
t
(6.28)
which occurs in the concatenation of the infinitesmal
propagator.
We may extend the integration limits of x by
defining H (x ) on the extended interval as H(IxI).
allows us to write E q . (6.28) as
This
51
f
O
2dq/ jr /
dx/2 sin(px)cos(q(x-a) ) e
—oo
(6.29)
since sin(qx)sin(qa) = cos(q(x-a)) - c o s (qx)cos(qa) and the
COs^ term must vanish as it gives an odd integrand.
We may-
further extend the limits of the 'momentum’ integration to
(-oo, oo) by extending H (q , I x I ) to H ( I q I
, IxI )
and writing
/ dxdq/ 2 jr sin(px) exp {iq(x-a) - iHAt }
for E q . (6.29).
(6.30)
We can do this for every pair of momentum
and position integrations except the last two in the
concatenation.
K (b I a )
Therefore we may write the propagator as
= / — co dc I0
x I
2 dp / jr e
( p ,c ) At
S£n (pb)sin(pc)
[dxdp/2jr] e xp {i/ px - H dt } .
(6.31)
The interior path integral is over all paths from a to c .
In the limit as At — > 0 we may perform the p integration to
give a pair of delta functions
K = / dc (S(b-c) - 8 (b + c )) KR (cla)
(6.32)
which gives back E q . (6.20).
The importance of this derivation is not only to show
how the method of delta functionals may be generalized but
also to show that one must always construct the propagator
from scratch on a new manifold.
been appreciated.
This fact has not always
For example Shulman (1981, p. 40) states
that the propagator on the half line would obtained by
simply restricting the position functional
the positive axis.
integrations to
As we have seen, this is incorrect.
52
CHAPTER 7
TIME-DEPENDENT COHERENT STATE PATH INTEGRALS
Recently,
tion operator)
coherent states
(eigenstates of the annihila­
for a time— dependent harmonic oscillator have
been discussed with the hope that these generalized coherent
states might be used to describe interactions between mole­
cules in lasers,
cf. Lewis and Riesenfeld (1968).
In
quantum field theory, coherent states are used to construct
path integrals which figure prominently in a functional
formalism,
cf . Itzykson and Zuber (1980) .
In particular,
a
path integral built from time-dependent coherent states
might be of value in discussing quantum field theories in
curved spacetime, cf. Berger (1982) and Ichinose
(1982).
Our interest in time—dependent coherent states is the fact
that they allow an easy evaluation of the generating func­
tional.
We calculate the generating functional for the time
—dependent harmonic oscillator at the end of the chapter.
We present the derivation of a time-dependent coherent
state path integral in this chapter.
We further show that
these states are eigenstates of an annihilation operator
that diagonaliz e s a new Hamiltonian obtained through a timedependent canonical transformation.
Hence this chapter is
an extension of the philosophy of Chapter 4 where we solved
53
path integrals through canonical transformations.
TIME-DEPENDENT COHERENT STATES
We first briefly discuss the properties of timedependent coherent states
of the path integral.
(TDCS) needed for the construction
The derivation of the path integral
is presented in the latter part of the chapter.
To begin, examine the Lagrangian
q2
2
<»2 (t)q2
2
(7.1)
and the associated Hamiltonian
-2
2 t ),2
H = —— + ----- .
(7.2)
2
2
One would normally diagonalize H by introducing operators a,
»
<
a+ and write q (t) and p (t ) as
q(t) = (a(t)+a+ (t))/(2w(t))1/2
(7.3)
p(t) = i(a+ (t)-a(t) ) (a)(t)/2)1/2,
(7.4)
but the Heisenberg equations of motion
H = to(t ) (a + a+1 / 2)
da"
= i [H , a+ ] +
At
dt
(7.5)
(7.6)
= ift)(t )a+ + “ - 7“T a
2ft)(t )
da
..„
,
d a+
.
ft) +
= i [H , a J + ---= — ift)a + — a
At
2 ft)
dt
lead to coupled equations of motion for a and a .
(7.7)
This
indicates a mixing of the positive and negative frequencies
or modes,
spoiling the interpretation of a and a+ as crea­
tion and annihilation operators.
More useful would be a description of q and p in terms
of operators that satisfy uncoupled equations of motion.
this end, one can write q and p in terms of s and y ,
To
54
functions to be determined.
q = (a+a+ )s (t ) ,
where y = I /2
Let
p = iy(a+-a)s + (a+a+ )s
(7.8)
.
One finds that the equations of motion for a, a+ can be
uncoupled if s (t ) satisfies the differential equation
d2s
2
I
+ (OS"" —“
= O e
dt
y)O3
As pointed out by Lewis and Riesenfeld
(7.9)
———
(1968), s(t) may be
any particular solution of this differential equation.
only condition we will make is that for w = constant,
— > (2a))"^ ^2 and y — > to.
The
s (t )
This enables us to compare our
results in the time-independent limit.
Notice that y as
defined here is the negative of that y discussed in Appendix
D.
Now a and a+ satisfy
a+ = iy(t)a+ , a = - iy(t )a .
(7.10)
The equation for s (t ) can be cast in a more suggestive form
by writing f(t) = s (t )e *
^.
Then f(t) satisfies
d2f(t)/dt + (o2 (t) f(t) = 0 ,
(7 .11 )
the differential equation for a harmonic oscillator with
time-dependent frequency (TDHO) .
The object of interest as regards path integrals is the
classical phase space action S , where
S = / (pq-pq) / 2 - H dt .
Expressing H in terms of a, a+
_ (a+a+ )2 -2
is
.7 7
v
(s — s"s") + —— (a+ — a ) + — (a+ a+aa + )
s
2
and noting
(7.12)
(7.13)
(7.14)
55
----- = -( aa+-aa+ ) + --(a+^-a ) + -(a+a+ ) (s^-ss)
2 . 2
s
2
one finds S to be
S = /
(a+ a-aa+ )/2i - (y/2)(a+ a+aa+ ) dt.
(7.15)
We define K = (y/2) (a + a+aa+ ) , I = K/y , and N = a+ a .
Classically and quantum mechanically, K is seen to be a new
Hamiltonian,
arrived at through a linear time-dependent
canonical transformation.
I is the much— discussed Lewis—
Riesenfeld invariant and N is a quantum mechanical number
operator.
One expects the transition amplitude from an initial
coherent eigenstate
I z £,>
to a final one <z £ I to have a path
integral representation
< Z£ Iz -> = f
[dz*dz/2jti] exp {iS[ z*, z] }
(7.16)
where the precise meaning of this symbolic notation will be
elaborated in the next section.
It will be shown that this
is indeed the case with one important caveat, to be
discussed.
TDCS can be constructed from the eigenstates of K .
these eigenstates be given by
Let
|n> , n = 0 , I, 2 , 3 ,... such
that
K In> =
y (n+1/2) In > .
.
The eigenvalue n is time-independent.
(7.17)
These eigenstates do
not evolve in time with the original Hamiltonian as the
generator of time displacements.
As shown by Lewis and
Riesenfeld, a state vector that does evolve via H can be
56
constructed by multiplying the eigenstates by a timedependent phase factor.
Specifically they find
In> s = eian (t)ln>
where
(7.18)
an satisfies the differential equation
don (t)
3
------ -- <n I i— — H (t ) I n > .
dt
91
In the case of the TDHO
we have introduced operators
a, a+ which have the properties
K = Y (a+ a+ l/ 2 ) ,
(7.19)
(7.20)
a+ I n> = (n+1 )■*•/^
I n+1 > ,
a I n> = U1^ ^ |n-i> .
If one fixes the phase of |0> by requiring
<-0 I ---|0> = — (ss"-s )
9t
2
so as to agree in the limit s — > (2to)~"* ^ ^ one finds
an (t) ---(n+1/2)Y .
(7.21)
(7.22)
Hence the Schroedinger state vector may be written as
I n (t )>s
= e-i7(a+l/2)|n,t> = e ' ^ O dt K ln.t>
(7.23)
where the time— dependence of the K eigenstates has been made
explicit.
We may now regard the
interaction state vectors
I n>
In,t> as 'rotating'
which evolve in time via the
new Hamiltonian K as observed from the 'rotating' frame of
reference.
Therefore we once again suppress the time— depen­
dence and write the state vectors as |n>.
Thus we have
implicitly defined a rotating reference frame in which K
generates time translations such that E q . (7.23) transforms
back to the standard Schroe dinger frame in which H generates
time translations.
As these vectors form a complete ortho-
normal set, any vector built from them will also evolve
57
according to K, in particular, the unnormaliz e d eigenstates
of a, Iz) where a Iz) = z|z).
The states
These are our coherent states.
Iz) can he formed by Iz) = eza I0>, the
eigenvalues z are time— independent and the states evolve
(as
seen from the rotating frame) according to
I z (t ))
= e i^O dt
K I z)
(7.24)
It is these TDCS Iz) that will be used to construct the
path integral which is the subject of the next section.
Notice that since the eigenvalues are time-independent one
has the important resolution of the identity
I = / dz*dz/2jti
I z) (z* I
e z z
(7 .25)
where dz*dz/2ni = d (Re z )d (Imz) / jr .
CONSTRUCTION OF THE PATH INTEGRAL
Coherent states were first used to construct a path
integral by Klauder (1960) and have now become part of the
standard texts.
Slavnov (1980)
We now follow the treatment of Faddev and
to present a brief derivation of the TDCS
path integral.
Any operator A has a kernel associated with it by
mapping it from Hilbert space into the complex numbers.
We
may write
(z* I A I z) =
2 (z * I n > <n IA |m > <m Iz)
(7.26)
and any two kernels A, B are convolved by
(7.27)
58
A normal symbol for A is a function A ^ (z*,z) found by
writing A as a function of a, a+ in normal order and then
letting the arguments a, a+
become complex c-numbers z and
z* respectively.
The relationship between kernel A and A^ can be shown
to be
A(z*,z)=e
A^fz*,z)
.
(7.28)
Let the new Hamiltonian K be given in normal order and
examine the kernel of the evolution operator U for short
times A t .
(7 .29)
U n U h-A1 .t> - <riK(z •z't)At
and the kernel is
U(t+At,t) = e
z ' z - i K (z * , z , t )At
(7.30)
Convolving N such kernels together such that NAt = t-r gives
*
U(z ,tIz ,r) = J
f
i
dzn dzn
zNzN-I
II ------ (e
2jti .
zI zO
x...xe
)
-Z^Z1
-iK(z^, Z ^ 1 )At
. - z1.N-I z N-I
z Izi1 ,
x (e
x . ..xe
) (e
—
xe
(zN - I »ZN_ 2 )At
-IK (Z 1 1 Z0 )At
x...xe
) .
(7.31)
Passing to the limit A t ---> 0, N --- > <», and symmetrizing
in Z1 and Z 1 one may write symbolically
*
.
— i f At K
(zf ,t|'z.,r) = ( zf Ie
I z j,) = U ( t , r )
(7.32)
= / [ - " " I exp {i/dt (—-——" — )-K( z , z)}exp ^(zfzf+z.z.)
which was the conjectured form.
Once again,
it is empha­
sized that this gives the propagator for the 'rotating'
states. This is the form for the propagator as seen from the
rotating frame of reference implicitly defined by E q .
(7.23) .
As a check, one can use this expression to find the
propagator for the position <b,t|a,r>.
Inserting complete
sets of TDCS one may write
,
v
, dz*dz, dz'dz.
-ZfZf -ZfZi
<b,tI a,r > = J — ---f
i e
e
2 TTI
2 JTI
where
x < b I z f( t ) ) ( z fIe
k I z ) ( z .(r ) I a >
i
i
<b I and I a > are position eigenstates.
(7.33)
Since K (z*,z) = y(t)(z*z + l/ 2) one has to evaluate a
Gaussian path integral which as we have seen simply has the
exponent evaluated along the classical path. This result
follows from the functional analogue of the method of
steepest descent or by concatenating the infinitesmal
propagators.
The classical equations of motion for z * and z
are
z *(t ) = iy(t)z*(t)
,
z (t ) = - iy(t )z (t )
(7.35)
with solutions
z * (t ) = Z^ e *^s
(7.34)
7 (t ) J
z (t ) = z^e-
^t Y (t )
The exponent becomes
(7.36)
- io(t ,r )/2
where c (t ,r ) = y (t ) - y (r ) hence the TDCS propagator
is
given by
(z^ ,11z f,r ) = exptZfZfe
- i a / 2}
(7 .3 7 )
60
The wavefunctions < x Iz) can be found by writing a in
terms of operator expressions for x and p in the position
representation and solving the resulting eigenvalue
differential equation.
<b I
Specifically
zfb
Jt -1/4
” zf
(t )) = (^--y)
exp{— -- + ---y - (Y (t )
(7.38)
s (t ) b '
- 7 ^~ 2 ^
— z *2
z ^a
s (r ) b
n -1/4
(z (r ) Ia > = (.---)
e x p {— -- + ---- - (y(r) + i----)--}
i
y(r)
2
s (r)
s(r) 2
so the propagator should be given by
K(bla) = (7r)'1/2 (y(t)y(r))1/4e" i<T(t'r)/2
(7.39)
.s(t)
b^
.
. s(r) a^
, dw*dw dz*dz
x exp{-(y(t)
s (r )
s (t) 2
2ni
.
.
r w2
wb
*
* -io
z* 2
z*a
*
x e xp {— — + ----— w w + w z e
— ---- + ----- — z z j
2
s (t )
2
s (r)
(7.40)
Using the formula E q . (B .7) one finds
1 ( 1 1 0 = L ~2 ni s incrTtTrT J
'
i
2•
26
x exp{--7— 7-- 7 ((b y(t)+a y (r ))
2 s inor (t , r )
x cosc(t,r)-2 (y(t)y(r))^/^ab)}
which is precisely the result given by Khandekar and
Lawande
(1975) .
The generating functional Z is found from the coherent
state path integral by letting z^ = z ^ = 0 and is
Z = exp{— io/2}.
(7.41)
In terms of an ancillary function $2 which is a particular
solution to
ift + $22
(7.42)
with limit $2 = to when w is a constant, Z may be written as
Z = exp{—
ixp {-
—J
$
J2(s)
2(s ) ds}.
(7.43)
61
CHAPTER 8
NEW SOLVABLE QUANTUM SYSTEM
In Chapter 4 we used the path integral solution to the
Hamiltonian
,2
P"
H -
(8 .1 )
to find the propagator for the inverse quadratic potential.
In this chapter we wish to present the complete solution to
the quantum mechanical problem posed by this Hamiltonian.
This solution is of interest because it is a non­
trivial generalization of the standard harmonic oscillator
which provides a new testing ground for the development of
functional methods which might be applied to more sophisti'
cated problems.
Also of interest is that the problem is the !— dimensio­
nal theory of non-linear scalar gravity and hence its
solution may provide some insight into questions concerning
the more complex 4— dimensional theory.
As we have seen this Hamiltonian is related to the
inverse quadratic potential, however no complete solution to
the inverse quadratic potential can be given as a generating
functional cannot be constructed thus preventing the develo­
pment of a quantum field-theoretical perturbation theory.
62
The solution to the classical problem is illuminating
because there are several analogous features in the quantum
solution.
Varying the action leads to the Euler— Lagrange
equations of motion which are
(D^ +
x = p (x^ + w^x^) /2 (l+|ix ) .
(8 .2 )
Since the Lagrangian is invariant under time transla­
tions the energy is conserved and takes the form
E = (x^ + w^x^)/2 (l+ (ix ) .
(8.3)
Therefore the equations of motion may be written as
(D^ + M^) x = |iE
with immediate solution
x (t ) = Ue- ^tot + a+ e^(ot + .(lEw-^ .
(8.4)
The particle oscillates about an equilibrium position X q =
pE<o~^ with amplitude A = (fi^E^w-^ + 2E(i)"*^)I/ 2 an^ frequency
to.
The quantum mechanic a I Hamiltonian associated with this
problem (with the correct ordering of operators)
r
H = (1+ nx )-^ ^p (1+ fix )■*■/^p (1+ nx )I / 2
+ to^x^/ 2 (l+ [ix ) -
is
(8.5)
3 2 (1+ jix ) .
This may be put into the more convenient form
H = p(l + fix)p/2 + M^x^/ 2 (1 + px ) .
(8 .6)
The quantum—mechanical Heisenberg equations of motion
are found to be
p = -HP^ / 2 - (2 + |ix )w^x (1+ fxx )- ^/2
(8.7)
x
(8 .8 )
and
( (1 + gx )p + p(l + |ix))/2
63
which may be combined with result
(D^ + ti)2)x = fiH,
(8.9)
the exact analogue of the classical equations.
the vacuum expectation value
We may take
(VEV) of this equation to find
(D^ + a)2)<x> = 6)2<x > = n<H> = fiE0 .
(8.10)
Therefore the VEV of the 'field' x (t ) is
<x> = pEgW ^ .
(8 .11)
This result will be used to check our construction of the
generating functional
in the next chapter.
The solution to E q . (8.9) is
x (t ) = Be " itot (2a))"1/2 + B+ e itot (2<o)-17 2 + jiHa)-2 .
(8.12)
The operators B and B+ are non— canonical creation and
annihilation operators and with the operator C = [B,B+ ] =
I + [I2H m -2 are easily found to form the Lie algebra of
S O (2,1)o
At this point it is also convenient to rescale the
operat ors .
Let
P — > (2w)l/2p, x — > x/(2(o)l/2 , p — > (2<i))*/ 2p . (8.13)
This canonical transformation allows us to write the
rescaled Hamiltonian as
H = to(p (1+px )p + x 2/ 4 (1 + px ) ) .
(8.14)
This Hamiltonian may be diagonaliz e d through the intro­
duction of canonical creation and annihilation operators.
These are defined via three successive sets of transforma­
t i ons
64
q. = x /2 (1+ jix ) + ip
(8.15)
q+ = x / 2 (1 + jix) - ip
X =
( q+q+ ) / (1-ji (q+q+ ) )
P = i( q+~q) /2
a = q/(1-jiq)
(8.16)
a+ = q+/ (I- Hq+)
q = a / (1+na)
q+ = a+/(1+H&+ )
A = a (I-H2 a+ a)-17 2
(8.17)
A+ = a+ (l-H2 aa+ )-172
a = A(1+ h 2A+A )-172
a+ = A+ (H-H2AA+)"172
The a 's are non-canonical creation and annihilation
operators with [a ,a+ ] ^ I and [H,a] = -wa,
[H, a+ ] = wa+ . The
canonical operators satisfy the standard commutation rela­
tions [A ,A+ ] = I and [H,A] = -wA,
[H ,A+ ] = m A „ The operators
B and B+ in terms of A and A+ are
B = A(1+A+A/ y )172
(8.19)
B+ = A+ (I+AA+ /y )17 2
(8.20)
where y = 2(i)h ~ 2 .
Using this result we find the commutator
[x „x ] = i(l+px)
and the time— dependent commutator
(8 .21 )
(8 .22 )
i[x(s),x(r)] = sin(wT) (! + Ji2H(I)- 2 ) /w - ji(x(s)-x(r)) m -2 .
The Hamiltonian may be written in terms of the non—
canonical operators as
65
M
a+ a
aa
(8.23)
^ l-(i^aa+
l-[i^a+ a
and in terms of the canonical operators as
H = to(AA+ + A+A) /2 = W (A+A + 1/2).
(8.24)
The spectrum may be determined with creation and
annihilation operators in the same manner as for the
harmonic oscillator and is found to be the same, namely En =
w(n+1/2) .
We may construct eigenstates
In> of a number operator N
= A+A with the properties
N In > = nIn>
(8.25)
H In > = M (n+1/2) |n>
(8.26)
A In> = (n)1/2 |n-l>
(8.27)
A+ In> = (n+l)1/2 ln+l>
(8.28)
Coherent states
(eigenstates of A) may be constructed
out of the ground state as follows
Normalized:
Iz> = e x p (z A+ - z *A)I0>, A Iz > = z Iz > (8.29)
Unnormalized:
Iz) = exp(zA+ ) I0 > , A Iz) = z Iz).
(8.30)
with Iz> = exp(~z*z/2 ) Iz) and
f d2 z/ JT IzXzI
= f d2 z/jr e-z z
I z) (z I
where d2 z/ Tt = dz*dz/2ni = dRezdlmz/n.
= I
(8.31)
The energy eigen­
functions <n I z) = z11/ (n I) "^"^ .
The spatial energy eigenfunctions <x I n > are most easily
found by examining the propagator from Chapter 4 which is
(in terms of the unrescaled variables)
66
K = ((l + (ib )(1+ jia
^ e
. 2 2
^ ' V-
(u / ijis in (wT/2) )
(8.32)
x e xp { (i(oc o t (uT/2) / fji^) ( (1+ fib ) + (1+ pa ) ) }
2 iw((1+ p b )(l+ p a ) ) ^
x I y l - --------------------- J .
p^sinCroT/2)
Using the Hille—Hardy formula E q . (B .1 3) we may rewrite the
propagator as
K=
2 e~ iu>T (n+1 / 2)
(2<o/ ^ ) ( (1 + pb ) (1+ pa ) )
x (n !/f (n+y+1 )) (e-11^2 uY^ 2 Ln^ (u) )
x (e” v/2 v?/2 LnY(y))
(8.33)
where Ln^ is the Laguerre polynomial and f the Gamma
function.
A glance at E q . (1.6) allows us to pick out the
spectrum as <o(n+1 / 2) and the eigenfunctions as
<u In> = (n! / f (n+y+1 ) ) -^2 (2 m u )
(8.34)
e-u ^ 2 Uy ^ 2 LnY (u)
and in terms of the rescaled variables u = (1+g x )/p2 and y =
1/fi2 .
We are now finished with the presentation of the exact
solution to the problem.
Hamiltonian,
We have found the spectrum of the
diagonalized it, and constructed the various
wavefunc tions and propagator.
The problem of calculating VEVs hinges upon the
construction of a generating functional Z [F] =
<0 ITexp (i/ F (t ),x (t ) ) I0> where T is the time— ordering symbol
placing all operators in chronological order.
the VEV <x(t)x(s)>
For example
(time— ordering will be implicitly under­
stood from now on) may be found by taking two functional
derivatives and two ordinary derivatives of Z [F] . For
67
example
<i(t)I<s,> - 7- ^ -
Z I F 1 Ie . O
( 8 -3 5 >
Now Z [F] in the presence of a potential or interaction
is hard to find but fortunately it may be written in terms
of Z o[F] the generating functional for a free field in the
presence of a source F(t).
The expression is
Z [F ] = exp(— i f V(8/ i&F(t )) dt)Zo[F],
(8.36)
Physically Z [F] represents the amplitude for the vacuum to
remain undisturbed in the presence of the force or source
F (t).
In quantum field theory that amplitude is taken to be
unity in the absence of sources requiring the normalization
Z [0] = I.
In Feynman diagram language this amounts to
neglecting all disconnected vacuum graphs or bubbles.
In this case we are actually able to solve for the
functional Z [F ] .
Using its definition as <0 10> in the
presence of sources we may write
Z [F ] = f f db da <0 Ib><b,sI a ,r><a I0>
(8.37)
in the limit as s and r go to plus and minus infinity.
Now <b,sla,r> is the propagator for the system driven by
some arbitrary time—dependent force F (t).
This was calcu­
lated in Chapter 4 and was found to be (unrescaled)
2
2
K = (2/ ijiD (s ,r ) ) ( (1 + pb) (1+ga) ) U ^ e ~ i f F/ji dt e i<o T/p
x exp{(2 i/p2D (s,r ))((1 + pb)D (s,r ) - (1 + pa)D (s »r ))}
x Iy {(-4i/p2D (s ,r ))((1+pb)(1+pa))^ 2 }
where (D2 + e2/4)D(s,r) = 8 (s ,r ) with
(8.38)
68
e^ (s ) = M ^ =
2{iF (s )
- 4|iwF(s)
(unrescaled)
x
(rescaled).
Using the previously given ground state wave functions
from Eq.
(8.34) and employing Eq.
(B.12) from Appendix B
enables us to perform the double integration giving the
result
(rescaled)
Z[F] =
dt e itoT(y+l)/2
x (-e-iwT/2 ---- eitoT/2D(s,r))™(Y +1) . '
to
dsdr
(8.39)
This result may be simplified by constructing an expli­
cit solution for D(s,r).
Define a new function fi by
ift + G2 = s2/4
(8.40)
with limit G = e/2 as e — > const.
Then by using the
identity Eq. (D.13) and Eq. (D.30) we may put this all
together to write as our final expression for the exact
generating functional
(rescaled)
Z[F] = exp(-i/ F/g) exp[-(y+l)Tr l n (8 - fiwFDc)]. (8.41)
69
CHAPTER 9
STRUCTURE OF THE GENERATING FUNCTIONAL
In this chapter we will compare the results obtained in
the last chapter with the corresponding results for the
harmonic oscillator (HO).
In particular, we wish to examine
the limit n — -> 0 closely to see if the limit is the naive
free HO or some pseudo— free HO (with perhaps a hard core).
W e will also derive the exact equations of motion formally
satisfied by the generating functional and see that
paradoxically they are not satisfied by Z .
curious result.
This is a very
Finally we discuss the perturbation theory
for the exact generating functional Z and introduce the last
new functional technique,
the concept of a functional anti-
derivative.
First we examine the limit of the propagator K to see
if it reduces to the HO result E q . (2.15).
The exact K is
K = - i ((l + [ia )(1+ pb ))■*•/^ e iwyT/Z (yy/2)^/2
x (sin(wT/2))~l exp {— cot (“ ) ( (1+ jib )+(1+ na ) ) }
x Iy {-iY ( (l + fib) (1 + na) )1/2 (sin(a)T/2)-:L} .
The limit n — > 0 is the same as y = Zmg-^ — > ” .
that Iy (— iz) = e
(9.1)
Noting
(z) and using the asymptotic limit
for J for large argument and order
70
Jy(y/cosp) ~ (2/ JTYtanp )
^ c o s (y t anp-yp-^n /4)
(9.2)
we find that
P = cos ^ {s in (ioT/2) (l+ |i(a+b )+ab
^^ }
(9.3)
and
tanp = (cos((oT/2)^+ n (a+b)+ a b j i ^ ) i n (m T / 2 ) .
(9.4)
Substituting these expressions into E q . (9.1) allows us to
write the limit for K in the form of a product of a pre­
exponential factor and a sum of exponential factors.
The limit as p — > 0 of the pre— exponential factors is
easily found to be
e-iji/4 (M /2jTisin(<oT))1 / 2 .
(9.5)
The exponential factors are found by expanding tanp — P to
order p2 .
After a great deal of algebra one finds
tanp-p ~ (toT-jt)/2 + co t (wT/2) (1+p (a+b )/ 2)
- g2 ( (b2 + a2 )cos (toT) - 2ab)/4sin(wT).
(9.6)
Substituting this into E q . (9.1) for K gives as the limit
K = K h o + KH 0 *exp {iy((i)T -
jt
+
2 c o t (wT/2 ) ) } .
(9.7)
In the limit y — > “ the second term on the right will
oscillate so rapidly relative to the first term that it will
fade out of existence relative to the first term.
that the term multiplying iy is strictly real.
Notice
Hence the
limit as p — > 0 of the propagator (and consequently the
wavefunctions) is the HO.
The wall at x = - 1/p moves off
to — 00 and the reflected (complex conjugate)
term fades away.
Now we wish to examine the limit for the generating
functional Z as given by E q . (8.41).
To do this it is best
71
to examine the structure of the generating functional.
We
introduce the concept of the connected generating functional
W which is just the logarithm of Z ,
W = In(Z).
(9.8)
In the case we are considering,
W = (-i/p)/ F - (H-Y)Tr In [8 - pFD/2] .
(9.9)
In the above and from here on out the Green's functions D
(and sometimes G) will he considered causal.
We also
introduce the functional M which is defined by
Mst...UlFl - Z-1 TJ5; JJ5J ••• JJ5J ZlE].
<9-1»)
When the source F is set to zero the functional M becomes
M 0 , an N-point Green's function <x(s)x(t) ...x(u)> which is a
VEV of a product of fields at different times.
One of the
primary objectives in quantum field theory is to calculate
these N-point Green's functions.
It is.for this reason that
the generating functional Z is introduced in the first
place.
To proceed further we also introduce one last
functional P
Pst....[Fl - J^5- -i-- ... - i -
WIFI.
(9.11)
We illustrate the relationship between the M's and Ps with
a few example s :
Ms = Ps,
. (9.12)
Mst = (PsPt + Pst),
(9.13)
Mstu = (PsPtPu + PtPsu + PuPst + PsPtu + Pstu).
(9.14)
A knowledge of the Ps enables one to construct the M's and
consequently the N—point Green's functions.
We now present
72
the general formula for P
Ps = -( I/g + in(1+ y )D ss/2)*
Ptl t2-'-tn = (|i/2i)n (l+r)
(9.15)
Dti t2Dt2 t3 • • -ntUtI •
(9.16)
Notice in the above formula for n > I that the sum is over
all (n-1) ! p e rmutations of the set 11 j »13 ,..., t } .
To
illustrate the use of this formalism and to check our
construction we calculate the VEV of the field x (t) which
was found in Chapter 8 to be (Eq. (8.11) with E q = m /2)
<x(t)> = n/2a).
(9.17)
Using our construction <x> is
<x(t)> = M 01 = P 01 = -( I/p + in(l+y)D*ss/2)
(9.18)
= - (I/ p + ip (l+2(0|i“^ ) (i/w )/ 2) = p/2<o .
(9.19)
The answers check.
We have now presented a complete analysis of the struc­
ture of the generating functional Z and have shown how to
calculate any quantity of interest.
We now return to the
discussion of the limit as p — > 0.
To show that the limit for Z is the HO we may show that
the limit for W is the HO result of
WH0 = f.
F tG°tsFs .
(9.20)
The Green's function G 01 s is the solution to the timeindependent harmonic oscillator differential equation
(D^ + (t)2)G0 t s = 8 ts .
(9.21)
Expanding the exact W (Eq. (9.9)) out to order p gives
W ~
F s (1+iMD°ss) +
Fs(D0St)^Ft + 0(p ).
(9.22)
R e m e m b e r that a superscript 0 indicates the F = O in the
73
functional.
Now we have the following relationships:
D 0St = ie-iwlTl/2 ,
M
(9.23)
D°ss = i/w,
(9.24)
G°st = -- e“ io)I T I = --(D°st)2 .
(9.25)
2o)
2i
Substituting the facts into the expression for W given by
E q . (9.19)
leads to the result
limit
— > 0, W — > WgQ.
Now we discuss an interesting paradox.
(9.26)
We have found
the correct generating functional Z and have examined its
limits.
We now wish to check whether it satisfies its
equation of motion.
An equation of motion for Z may be
derived from the functional integral for Z using the fact
that the integral of a derivative vanishes,
O = I
[dx]
therefore
(6/8x) ex p {iS} .
(9.27)
We may pull the functional derivative out of integral using
the fact that x = 8/ i8F giving
0 = iSS[x]/8x Z [F] with x = 8/18F.
(9.28)
Working out the derivative of the action leads to the
equation of motion for Z
0 = [- (1 + px )-l (D^+ti)2 )x + (p/2 ) (1 + nx )“^ (x^ + w^x^-fi^/16)
+ F + ip/2At (1 + fix )] Z [F ] .
(9.29)
The last term in the above sum arises from a correction that
must be added to the Lagrangi an which is iIn(l+px)/2At .
This creates an effective action which is used in the
equation of motion.
We may multiply the above equation by
(l+px)^ to write a new equation of motion
74
O = [-(l + (ix) (D^+w^) x + “ (x^ + <i>^x^ - " )
2
16
+ F (l+ [ix)^
(9.30)
— ——— (l+ |ix)] Z [F ]
2A t
Notice that [x,F] = - i/At. Working out the right-hand side
of the equation involves the addition of some 30 terms which
we omit from presenting.
When the dust clears,
one finds
that the right-hand side has the value
3H,((i) +
— —
+ — )
8
4
This is of course non-zero.
contradiction.
functional
satisfied.
(9.31)
Consequently we are left with a
Either we have calculated the generating
incorrectly or the equation of motion is not
We feel that the latter is the proper explana­
tion and present a simple argument for it.
By shifting the variables to p^z/4 = (l+px) one may put
the equation into the form
0 = [— z (D2 + 2£22) z/4 + z2/ 8 -3 iz/2At + g] Z'[F]
where Z ' is just Det- ^ +^
(9.32)
and g is the constant appearing
in the Lagrangian for the inverse quadratic potential g =
Y2/2 - 1/8.
Examine the above equation when Z' = Det^.
first three terms arise from the rescaled TDHO.
The
Consequen­
tly those terms operating upon Z ' must give a term like
CjB (l + 2p) since the roots for P are 0 and -1/2.
C cannot be a function of p.
The constant
Therefore E q . (9.32) when
evaluated must be of the form
2p2 + P + C g (p ) = 0.
When evaluated the constant C = -1/4.
(9.33)
The point is that the
P found from this method cannot satisfy the simple relation
-2
<x> = - 3 h /2 m - I/ n = fi/2o) = HE0(I)
(9.34)
if we demand as we must that C not be a function of p.
we have a paradox.
So
The formal equations of motion for Z
cannot be satisfied.
We now turn to an examination of the perturbation
theory for Z in terms of the harmonic oscillator.
Basically
the question we ask is if we did not know the exact Z what
information about it would we obtain through perturbation
theory.
In our ignorance of the exact solution we could
construct a perturbation expansion based upon the form (see
E q . (8.36)
in this regard)
„2
Z - e z p u / Ef(“ “
- P2 ) +
Z,>[F-J1
{I +|1X )
(9.35)
where x = 8/i8F and p = 8/ iSJ with
Z 0 = exp {
(9.36)
(I)2JG0J + FG0F - ZFG0Jj
and the Green's function G 0 defined as in E q . (9.21) and E q .
(9.25).
An examination of the above leads to some curious
conclusions.
solution.
First it does not look much like the exact
In fact from our knowledge of the exact solution
we know for example that the P 0Stu goes as
+ p which
implies that the perturbation series for P 0Stu in powers of
(I must somehow cut off at order
with all higher order
terms vanishing through some mechanism.
improbable.
This seems very
Second, we know that the exact P 0Stu involves
the Green's function D 0 which is in a real sense the square
root of G 0 .
It is difficult to imagine how products of G 0
76
can possibly create D 0 .
So just from a superficial
examination of the perturbative form for Z we expect little
if any information about the exact Z .
perturbation theory fails for Z .
In certain sense
To circumvent this we
create a new perturbation series for Z .
If we rescale L back to the inverse quadratic form of
E q . (4.23) we may write Z as
Z = e_i-f F/p exp {-ig/ (-4 i5/pSF)"1 } Det[F]-1Z2 .
This seems to be on the right track.
(9.37)
The first factor is
easily recognized to be present in the exact Z and the
determinant is' also present.
determinant is incorrect.
However the power of the
It was arrived at by pulling the
perturbative term out of the functional integral.
But we
know from the discussion in Chapter 4 that the limit g — > 0
is not the HO.
It is rather the HO with a reflecting wall .
Hence the determinant term which is the generating
functional for the TDHO must be replaced by the generating
functional for a TDHO with a reflecting wall.
This may be
worked out using the discussion in Chapter 6 and we rewrite
E q . (9.37) correctly as
Z = exp[-ZifT 2/ F} exp{|/ (6/5F)-1 } Det[F]-3/2
(9.38)
where for convenience in what is to follow we have rescaled
F — > 2F/fjL.
As we have remarked this perturbative form for
Z contains many features present in the exact Z .
lem now is to interpret the term (8/8F )—1 .
The prob­
77
In the theory of ordinary functions through the use of
Laplace transforms a term (d/dx)- * may be shown to be the
anti-derivative.
We appeal to this result to interpret the
similar term (8/SF)~* to be a functional anti-derivative.
Once this has been done we must now create a m e t h o d for
calculating the anti-derivatives of Det~^/2,
This may be
done by recognizing that the Fredholm determinant is a
linear functional of Ft,
ie: 5^/8Ft^ De t[F] = 0.
By intro­
ducing the projection operator (I-FsS/8F s ) Det = Det ^ )
and
the functional derivative notation 8/8Fs Det = De t ^g ^ we may
write the determinant symbolically as Det = Det^s^ +
FsDet^gJ.
The anti-derivative
(8/8F
s
Det^ is easily
found by writing
(8/SFs)—1 Detp = /Fs dS (Det(s) + SDet(s))P
= Detp+1/(p+l)Det(s).
(9.39)
Using the fact that Det ^ ^ = -Dss Det we may write the
above as
(8/SFs )~1 Detp = ( - ( p + D D s s )"1 Detp .
(9.40)
Preceding along the lines discussed we find that the second
functional anti— derivative is (for p = — 3/2)
(SZSFs)-I(SZSFt)-I Det_3/2 = -4Det“ 3/2 (Dst)-1
(9.41)
x (DssDtt - Dst2 )-1^2 Srctanf(DssDttDst-2 - I )1^2 } .
Beyond this we have not been able to explicitly compute the
anti-derivatives.
An important fact to note about the new
perturbation expansion is that it is an expansion in powers
of g which means an expansion in inverse powers of (if
78
CHAPTER 10
SUMMARY AND SPECULATION
In the proceeding chapters we have introduced three new
functional techniques.
delta functionals.
The principle one is the method of
By functionally integrating over the
position we showed how linear Hamiltonians gave rise to a
delta functional which constrained the momentum space paths
to obey an equation of evolution.
We were therefore able to
reduce the momentum functional integral to quadratures.
To augment our technique we developed three canonical
transformations which may he utilized in order to place the
Schroe dinger equation into a form where MDF might be
applied.
To illustrate each transformation, we worked a
physical example.
In the succeeding chapters we showed how one could
extend MDF to the relativistic case and to the situation
where the dynamics took place on a group manifold.
There
the momentum might take on discrete values or not even
exist.
In Chapter 7 we left our discussion of MDF and turned
to a new functional method,
the construction of a path
integral from time-dependent coherent states.
We then used
our new path integral to evaluate the generating functional
79
for the time— dependent harmonic oscillator.
We also checked
our path integral by deriving the known expression for the
TDHO propagator.
We presented another topic in Chapters 8 and 9.
There
we solved the one— dimensional analogue of general relativi­
ty.
AlI the techniques discussed earlier in this thesis
were brought to bear upon the construction of the generating
functional for this problem.
Then with the solution in hand
we discussed some of its peculiarities.
In particular we
showed that the generating functional appears not to satisfy
its formal equation of motion.
This is a very curious
result and could have some bearing upon other formal
functional equations.
We also discussed the formal perturbation theory for
our generating functional and argued that the theory seems
to produce nonsensical results.
A new perturbation theory
was constructed which utilized the concept of a functional
anti— derivative.
This concept is the last new functional
technique we developed.
We illustrated how to calculate
functional anti-derivatives and worked one example.
We also
presented the next formula in order of complexity.
Of the three methods presented in this thesis the
development of MDF is the most complete and consequently the
most exhausted.
There would appear to be little more MDF
could be applied to.
We have used it to solve the most
general problem it can be used to solve.
The creation of
80
MDF was a response to the tedious methods for explicitly
evaluating then available.
But its usefulness is limited.
One interesting result from its development is the bringing
to light of the common denominator that all (at present)
known calculable path integrals possess.
They may all be
placed into a form where the Hamiltonian is at most linear
in position.
The second technique,
that of the TDCS path integral
has more possible applications.
We have used it in particu­
lar to find the generating functional for the TDHO.
Other
possible uses are sketched in the opening paragraph of
Chapter 7.
An interesting result from our construction is
the recognition of the rotating frame of reference and its
role in simplifying one’s understanding of the quantum
dynamics of the TDHO and TDCS.
The new method that is the most interesting to us,
perhaps because it is the least developed of the three and
has yet to lose its mystery,
is the concept of the
functional anti— derivative introduced in the proceeding
chapter.
The concept of the antiderivative is important but
also important is the method introduced for calculating it.
In a sense it allows one to sum a certain set of Feynman
diagrams non-perturbatively.
The same method may be used to
evaluate a term like
exp{8/8F} De t [F]
(1 0 .1 )
#81
This would be of use in investigating Liouville field theory
where the Lagrangian density takes the form
L = (9x )2 - ex .
(10.2)
Liouv ille field theory is the higher dimensional analogue of
the Morse potential and has been the subject of recent
investigation in conjunction with string models for quarks.
Also the observation that the new perturbation theory
becomes an expansion in inverse powers of the coupling
constant is of potential
importance.
An obstacle in the
present theory of quantum chromodynamics is the large value
of the coupling constant rendering perturbation theory in
powers of the coupling constant suspect.
With the new
perturbation theory one might now have a way around this
difficulty
82
REFERENCES CITED
83
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86
APPENDICES
87
APPENDIX A
TIME-SLICING DERIVATIONS
)
88
In this appendix we derive equations
(1.18),
(3.11),
(1.4), (1.5),
(4.3) and (4.4) from our definition of the
time-sliced phase-space path integral of E q . (1.16) which we
write as
(A.I)
K(bla) = (II / dpn_/ 2 Jt) (II / dqn ) e 1 Se a ^ n - '4n .A t )
in the limit N ->
At -> 0 and where n+ = n+1/2. We do not
discuss whether the limit as so defined actually exists but
simply manipulate the formal time— sliced quantities in the
following.
The lattice action S ga is defined as
Sea = (^r
=
“ H(p,q) dt)ga
(A.2)
2 Pn-(qn - q.n_i) - l/2(H(pn_,qn ) + H (pn_ ,qn-1) )A t .
The Hamiltonian term is defined in such a way that variation
of the lattice action leads to the Euler approximation (ea)
3H
x
3H
to Hamilton's equations of motion p = - — (p,q), q = — (p ,q)
3q
dp
(Pn+ - pn - ) = - ™ 5 (^ " H(pn+'<ln) + ^ " H (pn- »In} } '
(A*3)
(qn " %n-l)
~ 5 (^Pn - <Pn" ,9n) + ^p' ? (Pn" ,qn" l) } * (A*4>
It is obvious that E q . (1.5) is satisfied by inspection of
Equations
(A.I) and
(A.2) .
The infinitesimal propagator
K i is
(A.5)
K i(b,c) = / dp/2jt exp(ip(b-c)
iAt (H(p ,b )+H(p ,c ) )/ 2)
and has the limit as At -> 0 (Eq. (1.4))
K i = / dp/2 jt eip(t-c) = 5(b-c).
(A.6)
To establish Eq. (1.18), Schroedinge r 's equation, we
first find the representation for the momentum p.
K (b ,T+At Ia ,0 ) = / dc K f (b , c )K (c ,T Ia ,0 )
Examine
(A.7)
89
(A.8)
and the limit as At — > O
Iim --(b,T+At) = / dpdc/2% (ip - i-- --(p,b)) E- E(cla)
0b
2 0b
f dp dc/2n (ip) e
^
0
K (c la) = --K(bla).
0b
(A.9)
This allows ns to identify the operator representation of p
as p = 3/ idb„
Eq.
(A.7) may be written as
K = / dc dp/ 2 it eip{b-c)
x ex p (-iAt(H(p ,b )+H(p,c))/2) K (cI a ) .
We may pull the exp(H)
(A.10)
terms out from inside the integral by
using the representation for p ..
To pull H (p ,b ) out we must
order all the b's to the left of the p's.
Then
K = exp (-iH (b Ip )A t / 2)
x / dc exp(-iH(p,c)At/2) 5(b - c ) K (cI a )
(A.11)
_m
n
= exp(-iH(b]p )A t /2) 2 H
— /dc — 5 (b-c) K (c la)
mn m !
nI
where we have expanded H in a Taylor series.
We may now
integrate with impunity with result
(A.12)
K(TH-At) = exp (-i(H(b |p)+H(p |b ) )At/2) K(T)
3K
Using — = Iim (K (T+At)—K(T))/At we may write
31
i--K(b,t I a ,r) = Hg(p,q) K (b » t I a ,r)
(A.13)
which was to be shown (Eq. (1.18)).
Next we demonstrate Eq.s (4.3) and (4.4)
fI
Notice that
dtea =
2
Pn-(9n-%n-l> = (PN-b -Pla) " 2 qn (pn+-pn_)
p q I - /r
dtea
which establishes Eq. (4.4).
/* h (q) q dt _ „ =
(A.14)
Examine
2 (h(qn )+h(qn_ 1 ))(qn- q n_ 1 )/2
(A.15)
90
and let dg/dq = h(q).
In the Euler approximation we may
write
(Sn-Sn-!) = (h(qn )+h( qn_1 ) ) C q ^ q nel)/2
(A.16)
and substituting this into E q . (A.15) gives
/® h(q)q dtea =
2 (gn-gn-l^ = S l =
s(b) - g(a)
(A.17)
which is E q . (4.3).
Finally we derive E q . (3.11).
Examine the functional
integral
K =
The
/[dp/2n][dq]
exp(i/®pq + qG(p) dt) H[p],
time-slicing representation is given by
K=
/ (II dpn_/27T)(II dqn ) H[pn ] e1 2 pn - ( qn"»n-l)
x e ^ ^ g (^n-) (qn+qn— I )^*/2
Here
(A.18)
(A.19)
NAt = s-r, q^ = q(r) = a, and q^ = q(s) = b.
Performing an integration by parts in the exponent lets
11
i
'
us write
(A.20)
<
K = / [dq/2 jt] [dp ]/ 2)r
exp(i/ q(p-G(p))
Noting that the position functional
dt) H[p].
integral is a
representation of a delta functional, we write
K = / [dp ] /2jr e ipql 6 [p-G(p)] H[p],
Using time-slicing Eq.s
(A.20) and
(A.21)
(A.21) are written as
K = / (II dpn_/ 2jr) (II dqn ) H [pn ] eib (pN- + g (PN-)At/2)
x e— ia(p1—— g (P1- )At/2)
x exp{-i 2 qn ((pn+-pn_)-(g(pn+ )+g(pn_))At/2)}
(A.22)
91
and
K = (Zn)-1Z d I
dpn HEpn ])
x e ib(pN+g(pN )At/2) Q-Ia(P1-S(P1)AtZZ)
x II 5 ( (pn+1-pn )-(g (pn+1)+ g (pn ))At/2)
where the + subscript has been dispensed with.
(A.23)
Notice that
the delta functions constrain p(t) to obey the numerical
Euler approximation to the differential equation
dp/dt = g(p), viz: Pn+i-pn = (g (Pn + i )+g(pn ))A t /2 .
It is desirable to collapse the concatenation of delta
functions by integrating
P^ integration,
from the bottom'
performed.
'down from the top'
then the p^_1 integration,
(performing the
etc.) and 'up
to leave a single pQ integration to be
Here to avoid confusion, read as 'p o h ’ rather
than 'p zero'.
To perform the integrations it is necessary
to express the argument of the delta function in terms of
the integrated variable.
a.
There are two cases:
5((pn+1-pn )-(g(pn+i)+g(pn ))At/2)
= 8(Pn+i-Pn-g(Pn )At) x (d(pn+1-g(pn+1)At/2)/dpn+1)-1
= 5(pn+l-pn-g(pn )At)/(1-g'(pn+1)At/2)
where the root
of pn+1-g(pn+1)A t/2 = pn+g(pn )At/2
(A.25)
is
written as Pn+1 = Pn+g(Pn )At.
Similarly
b.
8(...) = 8(pn+1-g(pn+1)At-pn )/ (1 + g '(pn )At/2)
(A. 26)
and to order At, these can be expressed as
a.
8 (pn+i-pn- g (Pn )At)(1+g'(pn + 1 )At/2)
b.
8 (pn+1-pn-g(pn+1)At)(l-g'(pn )At/2).
(A.27)
(A.28)
92
Integrating down from the top gives a product
II (1 + g '(pn )At/2) = II e g '(pn )At/2
and up from the bottom,
of
another
(A.29)
II e~g (pn )At/2 ^ith errors
At2 and every integration will force the momenta at each
time to become a recursive function of p0« Schematically: p^
--> Pn (P n -1) — > •••
P ^ ( P 0 ). We replace this by the
exact p(p0,T) (similarly for all pn ) with an error of At .
Now the propagator is written as
(A.30)
K = /dp 0 / 2 Jt H t p n (P0 )] II eg,(pm )At/2 x II e"g'(pm )At/2
x ei b (pN (p0 )+ g (pN )At/2) x e-ia(p1 (p0 )-g(p1)At/2)
In the limit as At— >0, N— >” , with NAt = s— r, displaying
the sums as Riemann integrals
K =
f_Z
dp0/2n H[p(p0 ,t)] e ibp(po ’s> x e- iap(po'r)
x e fI
g'(p)/2 dt x ,-7,
g '(p )/ 2 dte
(A.31)
Noting that
/x g'(p) dt = / dg(p)/dp dt
(A.32)
= / dg (dt/dp) = / dg/g = I n [g(s)/g (r )]
the last two factors can be expressed as
,g(s) g ( r ) l / 2
,
. .
^ d p (t )
(---- - - - rT)
and recognizing t h a t ----g (o ) g (o )
dp
the integral may be written as
ibp(s)-iap(r)
K
/_ dp0/2jr H[p(p0,t)]e
— g (p—)—dt
— 2% —»g—(t )
g (P0 ) dt
g (o)
(A.33)
d p (s ) dp(r) 1/2
dP o
dp0
where p(t) satisfies dp/dt = g(p), which was to be shown
(Eq.
(3.11)).
A few words are in order at this point.
First,
only
terms up to order At have been kept which is all that is
required for the concatenation to give the correct result.
93
Secondly,
in most instances the function g(p) will be
complex.
For a discussion of delta functions with complex
arguments see Richtmeyer (1978, p .3 8) .
The integration is
always over a real-valued p 0 .
Thirdly, for matrix equations of the form
p (t ) =
A (t ).p 0 delta functions of the type
8(A (t ).P 0 ) = 8(p0 )/detA
(A.34)
will be encountered so the requisite semigroup factors are
(detA(s)detA(r))^/^ with 'dp/dpQ ' = de tA, the Jacobian.
APPENDIX B
FORMULAE AND INTEGRALS
95
Table of Integrals and Formulae
f_l
(B .I )
dx xn e"ax2+bx = dn /dbn (7i/a)1/2 eb2/4a
f _ m ds dt e xp {-2 r st-(s-ib )2 - (t- ia)2 }
(B .2 )
= n d - r 2 )-1'2
1-r2
(B.3)
/_» ds dt exp(-2rst-as2-bt2 ) = n (ab-r2 ) 1 ^2
ds dt e xp (—2r st— as2 +2b s— c 12+2 dt)
(B .4 )
a(ac-r2 )-l/2 e x p { ^ - ^ - ^ ^ }
ac-r
The equations above are all special cases of the following
integral.
/
d^x exp(-x.A.x + b .x)
- (Jt)N/2 (de tA)
^ 2 exp {fe. A--*-,b/ 4 } .
(B .5 )
This integral admits a continuum or path integral generali­
zation
Next we list some formulae for complex Gaussian
integrals where the measure dz*dz/2ni = (dRez)(dlmz)/n .
f 11 dzn ‘dzn /2jti exp{-z*.A.z + a*e z + z*,b }
= (detA)-1 exp{a*.A- *.b } .
This integral also admits a continuum generalization.
(B „6)
96
In one dimension we have
f dz *dz / 2 jri exp(-az^ - bz*z - cz*^ + dz + ez *)
- (I2 - 4,«)-l/2
(B.7)
b
4 ac
/ dz*dz/2jti z *z exp (-z ^z + a*z + z*b)
= (I + a*b) exp(a*b).
(B .8)
Here is a concatenation formula for Gaussian integrals
ia ,(t-r)
f—
(r- s )
dr "p'-E<T;=ir + Ts^rn
-
expCii<*ii2 !,.
a(c-e)
0
.9 )
2(c-e)
f _ m dx x ^P+1^ exp{- --(x + r^/x)}
2
= -2jri r p Ip (- iar )0 (Re a ) .
In the above equation,
(B.10)
Ip is the modified Bessel
function of order p and 0 is the unit step function.
The
Bessel functions satisfy a Gaussian concatenation formula
.
/_
rdr e
2
Ip(-iar) !^(-ibr)
(B.ll)
a2+b 2
-- exp {— i (——-- )} I {-i " }
2a
4a
P
2a
Rep > - I , Re a > 0.
They also satisfy
f Q dxdy e ax by (xy)P^2 I ^ (-ic(xy)
2
- 4/
drds (Xs)P+1 e~as
2)
2
I(-icrs)
97
= 4 (-2 ic )** (4 at>+ c2 )_ (^+1) f(ji+l)
(B.12)
where f (x ) is the Gamma function.
The following formulae for bigene rating functions are useful
in determining wave functions from propagators.
The Hille-Hardy formula:
z
^
(1 — z) ^ exp {- — (x+y)
} I {-— (xyz ) ^ ^ }
(B .13 )
2
1— z
Ix I- z
= 2 zn n !/ f (n+n+1) e~ (x+y) / 2 (Xy)
Ln^ (x )Ln^ (y) .
Mehler's formula:
(l-z^ )
=
'i
^ eXp {y2 _ (y-ZX)2 / (]__z2 J J
2 zn Hn (x)Hn (y)/ (2nn ! ).
(B.14)
98
APPENDIX C
LINEAR MOMENTUM TRANSFORMATIONS
99
Finding the linear transformation that reduces a
quadratic Hamiltonian to one linear in position can he
troublesome so we list here some useful transformations for
standard Hamiltonians.
Time— dependent Harmonic Oscillator:
S = / pq “ a^(t )p^ /2 - b 2 (t )q^/2 - c(t)pq
+ d (t)p + F (t )q - E (t ) d t .
(C.l)
Let p = P— ifiQ, q = Q and define fi so that it is a
particular solution to iSi — b 2 — a2fi2 — 2 ic £2.
Then
S = (— i£2Q2 / 2 I + / Pd - a2 P2 / 2 - (c-ia2£2)PQ + Pd
+ (F-i£2d) Q - E dt .
(C.2)
Constant E and B Fields:
S = /
p .q - (p-mw(wxq))2Z2m + eq.E dt
(C.3)
with w = 1S and w = eB/2m.
Let p = P
- imti)(toxq) x m , q = Q, then
S = (— imw( wxCO 2Z 2 I + / P.6 - P2Zfm
+ toP. (toxQ+i (m x Q) xm) +e($. E dt .
(C .4 )
Rotating Coordinate System about ro:
S = / p.q - p2Z 2m + p.mxq d t .
(C .5 )
100
APPENDIX D
TIME-DEPENDENT HARMONIC OSCILLATOR
.I
101
In this appendix we collect several useful facts and
formulae concerning the time-dependent harmonic oscillator
(TDHO).
The TDHO is any system that obeys the differential
equation
(D2 + o>2 (t))f(t) = 0.
(D .I )
We may formally solve this equation in terms of the initial
data by constructing a Green's function that satisfies
(D2 + to2 )G(t,s) = 8(t ,s) .
(D .2 )
For the retarded Green's function we have the boundary
conditions Gjj(t.s) = 0 for t < s, Iim t — > s+ G^(t,s) = I,
and Gjj(t,t) = 0.
We may express f (t) in terms of f(s), £(s) at the
earlier time s by
f(t) = G jj(t ,s )f (s ) - GR (t,s)f(s)
(D .3 )
or using the notation 9" (see E q . (4.31))
f(t) = GR (t ,s)<Tsf (s ) .
(D.4)
The advanced Green's function G ^ (s,r ) = G R (r,s).
The advanced,
retarded,
and causal Green's functions
may be expanded in terms of the time— independent Green's
functions.
To do this let M2 (t ) = w2 - F (t ) and define G 0
as the solution to
(D2 + w2 )G0 = 8.
(D .5)
Then we may write G in terms of G0 as
G (s , r ) = G°(s,a)(8(a,r) - F (a )G<> (a ,r ) )_:L .
(D.6)
In the above equation we are using a continuous matrix
notation where repeated indices
(arguments of functions)
r
102
from the beginning of the alphabet are to be summed
(integrated) over.
To make this more obvious we will write
G (s,r ) as the matrix G sr.
A term like GsaGar stands for
/ G(s,a)G(a,r) da.
The variational derivative of G with respect to F is
8Gsr/8Ft = GstGtr.
(D .7)
It is important to note that the t in the above equation is
not summed over.
Summed indices are from the beginning of
the alphabet.
We may construct an explicit solution for G^ by
defining a new function £2(t ) by requiring it to be any
particular solution of
i& +
= w^ - F.
(D .8)
Then we may write
(D .9)
Gjjsr = O(s-r)/® dt e xp {i/* $2(u) du} e xp {i/* £2(v) dv} .
Notice that G is the product of a regular function TJ (the
integral)
and a distribution (the step function).
Using
this expression we find for the regular V
U
rs
r
=
— i£2sljRsr
+
exp {i/®
G
r
Sr
=
-iOrUgsr
-
e x p { - i/®
G
r
^
=
— £2s£2rG-R s r
+
i£2se —
The regular Wronskian is W =
Another useful
£2} ,
(D . I O )
£2} ,
^
—
(D . 1,1)
i£2re*^
(D . 1 2)
G^srUgsr - UgsrU^sf = I.
identity for Tj^ is
(D.1 3)
£2s£2rTjg,sr - i£2rGg,sr + i£2sUgsr + Ujjsr = -2 i£2r e
There is another explicit form for Gg that is used.
may be constructed by defining a new function N that is a
It
103
particular solution to the TDHO and writing N in the form
N = S(t )exp {iY (t) } .
Then 0 = iSf/N and
D2S - C2 S-3 + M2 S = 0,
(D .I4)
S2Y = C,
(D .15)
where C is some constant.
If we define another function
o (s ,r ) = Y (s ) - Y (r) we may write
as
Gjjsr = Osr s in (osr ) (y sy * ) ^ 2 .
Notice in the limit
Q
(D .I 6)
— > w and Y — > -to.
explicit form for G^ when M i s a
constant
Hence the
(G0) is
G 0 (T) = O(T)Sin(MT)Zw.
(D .1 7)
The function G^ may be canonic ally decomposed into
positive
(~ e
) and negative (~ e ^
) parts,
G r = G (+) + G (-) .
(D.I 8)
For example G 0R has the decomposition
Sin(MT)Zm = eit0^Z2iw - e- ^mtZ 2 im .
(D .I 9)
Here G ^+ ^ = ie~ ^ ^ Z 2m and G^- ^ = - ie ^to^Z 2m .
Notice that for
any w(t) we have
G (+)* = G (- ) ,
(D.20)
G^ + ^ sr = -G^ - ^rs,
(D .21 )
G^
(D .22)
ss + G^ ^ss = I.
The causal Green's function G q is constructed out of
these functions by
G(.sr = G^+ ^srO(s-r) - G^- ^srO( r-s ) .
For M = constant, G0c = iexp{-iw|T I} Z2w.
(D.23 )
Notice that the
causal or Feynman Green's function is symmetric,
= G q (r ,s) .
i e : G ^ (s,r )
104
The generalized Wronskian of
is
Wsr = G^srG^sr - G q SrG^s r = -5 srG^s s - 1/4.
(D .2 4)
We now wish to show that the function e x p { i f SI} used in
Eq.
(D .1 0) » etc. may be written as a continuous deter­
minant.
We do this so that we can explicitly display the
functional dependence of exp {i/ £2J upon F (t ) .
Once this is
done we may find its functional derivatives with respect to
F (t ) .
This is of importance because exp {i/ $2} arises in
connection with generating functionals of interest.
The generating functional of the TDHO was found in
Chapter 7 to be
Z [F] = exp {-
$2(t ) - w dt } .
Here s is strictly greater than r .
(D.25 )
In a time symmetric form
we may write Z as
Z [F 3 = exp{-
$2 - w} Os r + e xp { $ 2
- w JOr s .
(D .26)
However we may write Z as a path integral
Z = / [dx] exp {-
x (D^ + o)2 )x} .
(D. 27)
2
Using the continuum generalization of E q . (A2.5) allows us
to formally write Z as
Z = (Det [D2 + to(t)2] /DetED2 + w2 ])"1/2
(D.28)
= (Det[6 - F(t)G0c ts])—1 / 2 .
(D.29)
We have used the correct
(causal) boundary conditions to
pick out which Green's function should be used in the above
expression.
write
Hence we may equate E q s .
(D.26) and
(D.29) to
105
exp {i/ fi—w} 6 sr + exp {— i/ $2—w}0rs
(D.30)
= DetlS - FG0c ] .
The functional derivatives of the above expression
are
easily found to be
8/S F (t ) DetlS - FG 0c ] = -Ggtt DetIS - FG 0c ].
(D.31)
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